Level-Eight Eta Quotients: Theory & Applications

• Level-eight eta quotients are defined as eta products with exponents indexed by divisors {1,2,4,8}, featuring an explicit quadratic Nebentypus character and Newman’s congruence conditions.
• They exhibit well-structured transformation properties on groups like Γ₀(8) and Γ₁(8), with precise cusp-order tests ensuring holomorphy and modular behavior.
• Their arithmetic applications include modular parametrizations, Apéry-limit results linking to ζ(3), and identities via Rademacher sums and generalized double coset operators.

Searching arXiv for recent and relevant papers on level-8 eta quotients, Newman's theorem, and related modular-form applications. {"query":"level 8 eta quotient Newman theorem eta-quotient modular forms", "max_results": 10} A level-eight eta quotient is an eta quotient whose arguments are indexed by the divisors of $8$, namely

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$

with modular behavior governed by congruence conditions, cusp-order inequalities, and an explicit quadratic Nebentypus character. In the modern literature, level $8$ is a particularly tractable case: it appears in the elementary proof of Newman’s eta-quotient theorem, in counting and spanning problems for $\Gamma_0(8)$, in the classification of weight-$\tfrac12$ eta quotients, and in a level-$8$ modular parametrization of an Apéry-type differential equation leading to the identity $\lim B_n^{(8)}/s_n=(7/32)\zeta(3)$ (Savitt, 22 Jul 2025, Arnold-Roksandich et al., 2018, Shvets, 13 Apr 2026).

1. Definition and basic modular data

For level $8$, the divisor set is $\{1,2,4,8\}$, so every eta quotient has the form

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}.$$

Its weight is

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$0

Thus $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$1 if and only if

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$2

The level-$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$3 specialization of Newman’s congruence conditions is

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$4

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$5

In the Gordon–Hughes–Newman formulation on $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$6, the second condition is written equivalently as

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$7

At level $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$8, both congruences must be checked: they are independent in general (Savitt, 22 Jul 2025).

The Nebentypus is determined by

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$9

so for odd $8$0,

$8$1

Invariant	Level-$8$2 formula	Role
Eta quotient	$8$3	General form
Weight	$8$4	Integral weight condition
First congruence	$8$5	$8$6-invariance
Second congruence	$8$7	Conjugate $8$8-condition
Character	$8$9	Nebentypus

Two subgroup conventions coexist in the literature. Savitt’s presentation proves modularity on $\Gamma_0(8)$0 under Newman’s criterion, whereas the Gordon–Hughes–Newman theorem is stated on $\Gamma_0(8)$1 with Nebentypus $\Gamma_0(8)$2 (Savitt, 22 Jul 2025, Arnold-Roksandich et al., 2018).

2. Transformation theory and Newman’s criterion

The structural source of the level-$\Gamma_0(8)$3 congruences is the transformation theory of Dedekind’s eta function. The basic formulas are

$\Gamma_0(8)$4

For

$\Gamma_0(8)$5

Savitt proves

$\Gamma_0(8)$6

and

$\Gamma_0(8)$7

where

$\Gamma_0(8)$8

It follows that $\Gamma_0(8)$9 if and only if

$\tfrac12$0

and that

$\tfrac12$1

These are exactly the two level-$\tfrac12$2 Newman congruences (Savitt, 22 Jul 2025).

The sufficiency direction in Savitt’s proof has two ingredients. First, any integer-weight eta quotient is modular on the congruence subgroup

$\tfrac12$3

hence for $\tfrac12$4 on

$\tfrac12$5

with explicit Nebentypus

$\tfrac12$6

Second, a group-theoretic proposition shows that any congruence subgroup containing

$\tfrac12$7

contains $\tfrac12$8. This yields the $\tfrac12$9 modularity statement without appealing to Dedekind sums (Savitt, 22 Jul 2025).

The Gordon–Hughes–Newman theorem furnishes the parallel $8$0 formulation: if $8$1 and

$8$2

then

$8$3

for all $8$4, with

$8$5

(Arnold-Roksandich et al., 2018).

3. Cusp behavior and holomorphy

Weak modularity does not by itself imply holomorphy at the cusps. In Savitt’s $8$6 setting, the order criterion is stated as follows: for

$8$7

holomorphy at the cusps is necessary and sufficient when

$8$8

for all $8$9, with strict inequality for cusp forms. The level-$\lim B_n^{(8)}/s_n=(7/32)\zeta(3)$0 specialization gives the four inequalities

$\lim B_n^{(8)}/s_n=(7/32)\zeta(3)$1

$\lim B_n^{(8)}/s_n=(7/32)\zeta(3)$2

$\lim B_n^{(8)}/s_n=(7/32)\zeta(3)$3

$\lim B_n^{(8)}/s_n=(7/32)\zeta(3)$4

Strict positivity in all four inequalities gives cusp forms on $\lim B_n^{(8)}/s_n=(7/32)\zeta(3)$5 (Savitt, 22 Jul 2025).

In the $\lim B_n^{(8)}/s_n=(7/32)\zeta(3)$6 counting framework, the cusp order at denominator $\lim B_n^{(8)}/s_n=(7/32)\zeta(3)$7 is

$\lim B_n^{(8)}/s_n=(7/32)\zeta(3)$8

Explicitly,

$\lim B_n^{(8)}/s_n=(7/32)\zeta(3)$9

$8$0

$8$1

$8$2

Holomorphicity on $8$3 requires

$8$4

and cusp-form status requires all four to be $8$5 (Arnold-Roksandich et al., 2018).

These two sets of formulas are presented for different modular groups, respectively $8$6 and $8$7. A plausible implication is that level $8$8 is unusually convenient precisely because both subgroup settings admit explicit linear cusp-order tests.

4. Distinguished level-$8$9 constructions

Several level-$\{1,2,4,8\}$0 eta quotients play canonical roles.

A central pair is

$\{1,2,4,8\}$1

Their Fourier expansions begin

$\{1,2,4,8\}$2

$\{1,2,4,8\}$3

They satisfy

$\{1,2,4,8\}$4

at the cusps $\{1,2,4,8\}$5 of $\{1,2,4,8\}$6. Under the Fricke involution

$\{1,2,4,8\}$7

one has

$\{1,2,4,8\}$8

Because $\{1,2,4,8\}$9 has genus $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}.$$0, $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}.$$1 is a Hauptmodul for $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}.$$2, and the modular parametrization

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}.$$3

encodes the holomorphic solution of a third-order Picard–Fuchs equation (Shvets, 13 Apr 2026).

A second explicit family, obtained in the $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}.$$4 construction with $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}.$$5, is

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}.$$6

with character

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}.$$7

Its exponent vector is

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}.$$8

The two Gordon–Hughes–Newman congruences are satisfied for all integers $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}.$$9, and its cusp orders obey

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$00

with $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$01. Thus it is holomorphic at all cusps but not cuspidal at $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$02. For $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$03,

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$04

This gives an infinite family of noncuspidal modular eta quotients on $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$05 (Arnold-Roksandich et al., 2018).

A third structural object is the canonical eta-quotient Hauptmodul $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$06 for $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$07. Its exact eta-product expression is not printed in the provided excerpt, but its Fourier coefficients are given by a Rademacher sum: $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$08 where $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$09 is defined through modified Bessel functions and Kloosterman-type sums. This places level $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$10 among the genus-$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$11 levels

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$12

for which eta-quotient Hauptmoduln admit uniform Rademacher expansions (Acres et al., 2018).

5. Classification, counting, and multiplier systems

Level $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$13 occupies several sharp classification thresholds.

For weight $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$14, every holomorphic eta quotient is an integral rescaling of one of the fourteen primitive eta quotients in Zagier’s list. No primitive member of that list has level $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$15. Consequently, every holomorphic eta quotient of weight $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$16 and level $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$17 is a rescaling of a primitive eta quotient of level $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$18, $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$19, or $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$20, and there does not exist any simple holomorphic eta quotient of level $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$21 (Bhattacharya, 2016).

From the irreducibility theory of holomorphic eta quotients, the level-$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$22 bound is especially small. For

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$23

one has

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$24

Therefore any irreducible holomorphic eta quotient of level $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$25 has weight $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$26, and there are only finitely many irreducible holomorphic eta quotients of level $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$27 (Bhattacharya, 2016).

At the level of enumeration, the genus-zero counting theorem gives the exact number of holomorphic eta quotients in $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$28: $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$29 for each even weight $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$30. The same work proves

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$31

and therefore, for all even $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$32,

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$33

Thus every holomorphic form in $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$34 is a linear combination of weakly holomorphic level-$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$35 eta quotients with poles only at $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$36 (Rouse et al., 2013).

The multiplier-system classification is equally explicit. For integral-exponent eta quotients on the double cover $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$37, every fixed weight $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$38 admits exactly $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$39 distinct eta-quotient multiplier systems. A complete set of representatives is given by exponent patterns

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$40

with

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$41

For integral $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$42, these descend to characters on $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$43 itself (Zhu, 2024).

A further small-weight phenomenon appears in the operator theory of generalized double cosets. Since

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$44

every space $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$45 is at most one-dimensional. Table 4.3 of the cited work lists the level-$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$46, weight-$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$47 holomorphic eta quotients explicitly and uses this one-dimensionality to show that compatible generalized Hecke operators $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$48 must send one such eta quotient to a scalar multiple of another (Zhou et al., 2021).

6. Arithmetic and analytic applications

The most striking arithmetic application presently attached to level $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$49 is the Apéry-limit theorem. Define

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$50

and let $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$51 be the rational companion sequence satisfying the same recurrence as $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$52, with generating series $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$53. The modular parametrization

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$54

and the Eichler-integral quotient

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$55

lead to the weight-$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$56 modular form

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$57

Its $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$58-function satisfies

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$59

and the Fricke period polynomial identity becomes

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$60

From the dominant critical value

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$61

one obtains the limit

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$62

and hence the continued-fraction identity

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$63

This is the level-$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$64 Apéry limit and the proof of the Ramanujan Machine conjecture Z1 (Shvets, 13 Apr 2026).

Level $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$65 also supports explicit coefficient decompositions arising from generalized double coset operators. Among the identities obtained are

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$66

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$67

and the four congruence-class decompositions

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$68

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$69

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$70

$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$71

These identities express residue-class subsequences of eta-power coefficients in terms of level-$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$72 eta quotients (Zhou et al., 2021).

Several misconceptions are excluded by the current theory. First, level $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$73 does not imply novelty: at weight $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$74, there is no primitive simple level-$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$75 object. Second, weak modularity is not the same as cusp holomorphy; the linear cusp-order constraints remain essential. Third, at level $$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$76 the two Newman congruences are genuinely independent. Precisely because these issues are now explicit, level-$$f(z)=\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}, \qquad r_1,r_2,r_4,r_8\in\mathbb Z,$$77 eta quotients serve as a model case in which transformation laws, cusp geometry, multiplier systems, irreducibility, and arithmetic applications can all be written down in concrete form.