---
title: 'Level-Eight Eta Quotients: Theory & Applications'
url: https://www.emergentmind.com/topics/level-eight-eta-quotient
type: topic
---

# Level-Eight Eta Quotients: Theory & Applications

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 \(\lim B_n^{(8)}/s_n=(7/32)\zeta(3)\) [2507.16225], [1811.07244], [2604.14219].

## 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
\[
k=\frac12(r_1+r_2+r_4+r_8).
\]
Thus \(k\in\mathbb Z\) if and only if
\[
r_1+r_2+r_4+r_8\equiv 0\pmod 2.
\]

The level-\(8\) specialization of Newman’s congruence conditions is
\[
r_1+2r_2+4r_4+8r_8\equiv 0\pmod{24},
\]
\[
r_8+2r_4+4r_2+8r_1\equiv 0\pmod{24}.
\]
In the Gordon–Hughes–Newman formulation on \(\Gamma_0(8)\), the second condition is written equivalently as
\[
8r_1+4r_2+2r_4+r_8\equiv 0\pmod{24}.
\]
At level \(8\), both congruences must be checked: they are independent in general [2507.16225].

The Nebentypus is determined by
\[
\chi(d)=\left(\frac{(-1)^k s}{d}\right),
\qquad
s=1^{r_1}2^{r_2}4^{r_4}8^{r_8}=2^{\,r_2+2r_4+3r_8},
\]
so for odd \(d\),
\[
\chi(d)=\left(\frac{(-1)^k2^{\,r_2+2r_4+3r_8}}{d}\right).
\]

| Invariant | Level-\(8\) formula | Role |
|---|---|---|
| Eta quotient | \(\eta(z)^{r_1}\eta(2z)^{r_2}\eta(4z)^{r_4}\eta(8z)^{r_8}\) | General form |
| Weight | \(k=\frac12(r_1+r_2+r_4+r_8)\) | Integral weight condition |
| First congruence | \(r_1+2r_2+4r_4+8r_8\equiv0\pmod{24}\) | \(T\)-invariance |
| Second congruence | \(r_8+2r_4+4r_2+8r_1\equiv0\pmod{24}\) | Conjugate \(T\)-condition |
| Character | \(\chi(d)=\left(\frac{(-1)^k2^{r_2+2r_4+3r_8}}{d}\right)\) | Nebentypus |

Two subgroup conventions coexist in the literature. Savitt’s presentation proves modularity on \(\Gamma_1(8)\) under Newman’s criterion, whereas the Gordon–Hughes–Newman theorem is stated on \(\Gamma_0(8)\) with Nebentypus \(\chi\) [2507.16225], [1811.07244].

## 2. Transformation theory and Newman’s criterion

The structural source of the level-\(8\) congruences is the transformation theory of Dedekind’s eta function. The basic formulas are
\[
\eta(z+1)=e^{\pi i/12}\eta(z),\qquad
\eta(-1/z)=e^{-\pi i/4}\sqrt{z}\,\eta(z).
\]
For
\[
f(z)=\prod_{m\mid 8}\eta(mz)^{r_m},
\qquad
k=\frac12\sum_{m\mid 8}r_m,
\qquad
s=\prod_{m\mid 8}m^{r_m},
\]
Savitt proves
\[
f|_T=e^{\pi i b/12}f,
\qquad
b=\sum_{m\mid 8}mr_m,
\]
and
\[
f|_{w_8}=e^{-\pi i k/2}\sqrt{s}\,f^*,
\qquad
w_8=\begin{pmatrix}0&-1\\8&0\end{pmatrix},
\]
where
\[
f^*(z)=\prod_{m\mid 8}\eta(mz)^{r_{8/m}}.
\]
It follows that \(f|_T=f\) if and only if
\[
r_1+2r_2+4r_4+8r_8\equiv0\pmod{24},
\]
and that
\[
f\Big|_{\begin{pmatrix}1&0\\8&1\end{pmatrix}}=f
\iff
f^*|_T=f^*
\iff
r_8+2r_4+4r_2+8r_1\equiv0\pmod{24}.
\]
These are exactly the two level-\(8\) Newman congruences [2507.16225].

The sufficiency direction in Savitt’s proof has two ingredients. First, any integer-weight eta quotient is modular on the congruence subgroup
\[
\Gamma_0(24N)\cap\Gamma^0(24)\cap\Gamma(4N),
\]
hence for \(N=8\) on
\[
\Gamma_0(192)\cap\Gamma^0(24)\cap\Gamma(32),
\]
with explicit Nebentypus
\[
\left(\frac{(-1)^k s}{d}\right).
\]
Second, a group-theoretic proposition shows that any congruence subgroup containing
\[
T=\begin{pmatrix}1&1\\0&1\end{pmatrix}
\quad\text{and}\quad
\begin{pmatrix}1&0\\8&1\end{pmatrix}
\]
contains \(\Gamma_1(8)\). This yields the \(\Gamma_1(8)\) modularity statement without appealing to Dedekind sums [2507.16225].

The Gordon–Hughes–Newman theorem furnishes the parallel \(\Gamma_0(8)\) formulation: if \(k\in\mathbb Z\) and
\[
\sum_{\delta\mid 8}\delta r_\delta\equiv0\pmod{24},
\qquad
\sum_{\delta\mid 8}\frac{8}{\delta}r_\delta\equiv0\pmod{24},
\]
then
\[
f|_k\gamma=\chi(d)f
\]
for all \(\gamma\in\Gamma_0(8)\), with
\[
\chi(n)=\left(\frac{(-1)^k\,2^{\,r_2+2r_4+3r_8}}{n}\right)
\]
[1811.07244].

## 3. Cusp behavior and holomorphy

Weak modularity does not by itself imply holomorphy at the cusps. In Savitt’s \(\Gamma_1(8)\) setting, the order criterion is stated as follows: for
\[
f(z)=\prod_{m\mid 8}\eta(mz)^{r_m},
\]
holomorphy at the cusps is necessary and sufficient when
\[
\sum_{m\mid 8}\frac{(c,m)^2}{m}r_m\ge 0
\]
for all \(c\mid 8\), with strict inequality for cusp forms. The level-\(8\) specialization gives the four inequalities
\[
r_1\ge 0,
\]
\[
r_1+2r_2+r_4+\tfrac12 r_8\ge 0,
\]
\[
r_1+2r_2+4r_4+2r_8\ge 0,
\]
\[
r_1+2r_2+4r_4+8r_8\ge 0.
\]
Strict positivity in all four inequalities gives cusp forms on \(\Gamma_1(8)\) [2507.16225].

In the \(\Gamma_0(8)\) counting framework, the cusp order at denominator \(d\mid 8\) is
\[
v_d=\frac{8}{24}\sum_{\delta\mid 8}\frac{(d,\delta)^2}{(d,8/d)\,d\,\delta}\,r_\delta
=\frac13\sum_{\delta\mid 8}\frac{(d,\delta)^2}{(d,8/d)\,d\,\delta}\,r_\delta.
\]
Explicitly,
\[
v_1=\frac13\Bigl(r_1+\tfrac12 r_2+\tfrac14 r_4+\tfrac18 r_8\Bigr),
\]
\[
v_2=\frac13\Bigl(\tfrac14 r_1+\tfrac12 r_2+\tfrac14 r_4+\tfrac18 r_8\Bigr),
\]
\[
v_4=\frac13\Bigl(\tfrac18 r_1+\tfrac14 r_2+\tfrac12 r_4+\tfrac14 r_8\Bigr),
\]
\[
v_8=\frac13\Bigl(\tfrac18 r_1+\tfrac14 r_2+\tfrac12 r_4+r_8\Bigr).
\]
Holomorphicity on \(\Gamma_0(8)\) requires
\[
v_1,v_2,v_4,v_8\ge 0,
\]
and cusp-form status requires all four to be \(>0\) [1811.07244].

These two sets of formulas are presented for different modular groups, respectively \(\Gamma_1(8)\) and \(\Gamma_0(8)\). A plausible implication is that level \(8\) is unusually convenient precisely because both subgroup settings admit explicit linear cusp-order tests.

## 4. Distinguished level-\(8\) constructions

Several level-\(8\) eta quotients play canonical roles.

A central pair is
\[
t(\tau)=\frac{\eta(\tau)^8\eta(8\tau)^8}{\eta(2\tau)^8\eta(4\tau)^8},
\qquad
Y(\tau)=\frac{\eta(2\tau)^6\eta(4\tau)^6}{\eta(\tau)^4\eta(8\tau)^4}.
\]
Their Fourier expansions begin
\[
t(\tau)=q-8q^2+28q^3-64q^4+142q^5-352q^6+O(q^7),
\]
\[
Y(\tau)=1+4q+8q^2+16q^3+24q^4+24q^5+32q^6+O(q^7).
\]
They satisfy
\[
\ord(t)=(1,1,-1,-1),
\qquad
\ord(Y)=(0,0,1,1)
\]
at the cusps \(\infty,0,1/2,1/4\) of \(\Gamma_0(8)\). Under the Fricke involution
\[
W_8(\tau)=-\frac{1}{8\tau},
\]
one has
\[
t(W_8\tau)=t(\tau),
\qquad
Y(W_8\tau)=-8\tau^2Y(\tau),
\qquad
Y|_2W_8=-Y.
\]
Because \(X_0(8)^+\) has genus \(0\), \(t\) is a Hauptmodul for \(\Gamma_0(8)^+\), and the modular parametrization
\[
Y(\tau)=\mathcal A(t(\tau))
\]
encodes the holomorphic solution of a third-order Picard–Fuchs equation [2604.14219].

A second explicit family, obtained in the \(4p\) construction with \(p=2\), is
\[
f_k(z)=\frac{\eta(8z)^{4k}}{\eta(4z)^{2k}}\in M_k(8,\chi),
\qquad k\ge 0,
\]
with character
\[
\chi(n)=\left(\frac{(-1)^k}{n}\right).
\]
Its exponent vector is
\[
r_1=0,\qquad r_2=0,\qquad r_4=-2k,\qquad r_8=4k.
\]
The two Gordon–Hughes–Newman congruences are satisfied for all integers \(k\), and its cusp orders obey
\[
v_1=0,\qquad v_8=k,
\]
with \(v_2,v_4\ge 0\). Thus it is holomorphic at all cusps but not cuspidal at \(\infty\). For \(k=1\),
\[
\frac{\eta(8z)^4}{\eta(4z)^2}=q+O(q^5).
\]
This gives an infinite family of noncuspidal modular eta quotients on \(\Gamma_0(8)\) [1811.07244].

A third structural object is the canonical eta-quotient Hauptmodul \(T_8/B_8\) for \(\Gamma_0(8)\). Its exact eta-product expression is not printed in the provided excerpt, but its Fourier coefficients are given by a Rademacher sum:
\[
\frac{T_8}{B_8}(\tau)=q^{-1}+\sum_{n\ge 0}\frac{R_{8,1}(n)}{R_{8,1}(1)}q^n,
\]
where \(R_{8,1}(n)\) is defined through modified Bessel functions and Kloosterman-type sums. This places level \(8\) among the genus-\(0\) levels
\[
N=1,2,3,4,5,6,7,8,9,10,12,13,16,18,25
\]
for which eta-quotient Hauptmoduln admit uniform Rademacher expansions [1810.07478].

## 5. Classification, counting, and multiplier systems

Level \(8\) occupies several sharp classification thresholds.

For weight \(\tfrac12\), 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 \(8\). Consequently, every holomorphic eta quotient of weight \(\tfrac12\) and level \(8\) is a rescaling of a primitive eta quotient of level \(1\), \(2\), or \(4\), and there does not exist any simple holomorphic eta quotient of level \(8\) [1602.02835].

From the irreducibility theory of holomorphic eta quotients, the level-\(8\) bound is especially small. For
\[
\kappa(N)=\varphi(\mathrm{rad}(N))\prod_{p^n\parallel N}\big((n-1)(p-1)+2\big),
\]
one has
\[
\kappa(8)=4,
\qquad
\kappa(8)/2=2.
\]
Therefore any irreducible holomorphic eta quotient of level \(8\) has weight \(<2\), and there are only finitely many irreducible holomorphic eta quotients of level \(8\) [1602.02814].

At the level of enumeration, the genus-zero counting theorem gives the exact number of holomorphic eta quotients in \(M_k(\Gamma_0(8))\):
\[
\frac{k^3}{6}+k^2+\frac{11k}{6}+1
\]
for each even weight \(k\). The same work proves
\[
\mathcal R_2(\Gamma_0(8))=\mathcal E_2(\Gamma_0(8)),
\]
and therefore, for all even \(k\ge0\),
\[
\mathcal R_k(\Gamma_0(8))=\mathcal E_k(\Gamma_0(8)).
\]
Thus every holomorphic form in \(M_k(\Gamma_0(8))\) is a linear combination of weakly holomorphic level-\(8\) eta quotients with poles only at \(\infty\) [1311.1460].

The multiplier-system classification is equally explicit. For integral-exponent eta quotients on the double cover \(\widetilde{\Gamma_0(8)}\), every fixed weight \(k\in\tfrac12\mathbb Z\) admits exactly \(384\) distinct eta-quotient multiplier systems. A complete set of representatives is given by exponent patterns
\[
\mathbf r=2k\,e_1+r_2e_2+r_4e_4+r_8e_8,
\]
with
\[
r_2\in\{0,1\},\qquad 0\le r_4<8,\qquad 0\le r_8<24.
\]
For integral \(k\), these descend to characters on \(\Gamma_0(8)\) itself [2408.00246].

A further small-weight phenomenon appears in the operator theory of generalized double cosets. Since
\[
[PSL_2(\mathbb Z):\overline{\Gamma_0(8)}]\cdot \tfrac12<12,
\]
every space \(M_{1/2}(\widetilde{\Gamma_0(8)},v)\) is at most one-dimensional. Table 4.3 of the cited work lists the level-\(8\), weight-\(\tfrac12\) holomorphic eta quotients explicitly and uses this one-dimensionality to show that compatible generalized Hecke operators \(T_l\) must send one such eta quotient to a scalar multiple of another [2110.06768].

## 6. Arithmetic and analytic applications

The most striking arithmetic application presently attached to level \(8\) is the Apéry-limit theorem. Define
\[
s_n=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}{n}^2,
\qquad
\mathcal A(z)=\sum_{n\ge0}s_n z^n,
\]
and let \(B_n^{(8)}\) be the rational companion sequence satisfying the same recurrence as \(s_n\), with generating series \(\mathcal B(z)\). The modular parametrization
\[
Y(\tau)=\mathcal A(t(\tau))
\]
and the Eichler-integral quotient
\[
E(\tau)=\frac{\mathcal B(t(\tau))}{Y(\tau)}
\]
lead to the weight-\(4\) modular form
\[
g_8(\tau)=\frac{E_4(\tau)-21E_4(2\tau)+84E_4(4\tau)-64E_4(8\tau)}{240}.
\]
Its \(L\)-function satisfies
\[
L(g_8,3)=\frac{7}{32}\zeta(3),
\]
and the Fricke period polynomial identity becomes
\[
(E|_{-2}W_8)(\tau)+E(\tau)=\frac{7}{32}\zeta(3)\,(8\tau^2+1).
\]
From the dominant critical value
\[
t_0=\frac{3-2\sqrt2}{4},
\]
one obtains the limit
\[
\lim_{n\to\infty}\frac{B_n^{(8)}}{s_n}=\frac{7}{32}\zeta(3),
\]
and hence the continued-fraction identity
\[
\operatorname{PCF}\bigl((2n+1)(3n^2+3n+1),-n^6\bigr)=\frac{8}{7\zeta(3)}.
\]
This is the level-\(8\) Apéry limit and the proof of the Ramanujan Machine conjecture Z1 [2604.14219].

Level \(8\) also supports explicit coefficient decompositions arising from generalized double coset operators. Among the identities obtained are
\[
\eta(\tau)\eta(2\tau)^{-2}\eta(4\tau)^5\eta(8\tau)^{-2}
= q^{11/24}\sum_{n\ge0}P_2(2n)q^n,
\]
\[
\eta(\tau)\eta(4\tau)^{-1}\eta(8\tau)^2
= -\tfrac12 q^{13/24}\sum_{n\ge0}P_2(2n+1)q^n,
\]
and the four congruence-class decompositions
\[
\eta(\tau)^5\eta(2\tau)^{-4}\eta(4\tau)^5\eta(8\tau)^{-2}
= q^{1/24}\sum_{n\ge0}P_4(4n)q^n,
\]
\[
\eta(\tau)^{-1}\eta(2\tau)^2\eta(4\tau)^5\eta(8\tau)^{-2}
= -\tfrac14 q^{7/24}\sum_{n\ge0}P_4(4n+1)q^n,
\]
\[
\eta(\tau)^5\eta(2\tau)^{-2}\eta(4\tau)^{-1}\eta(8\tau)^2
= \tfrac12 q^{13/24}\sum_{n\ge0}P_4(4n+2)q^n,
\]
\[
\eta(\tau)^{-1}\eta(2\tau)^4\eta(4\tau)^{-1}\eta(8\tau)^2
= \tfrac18 q^{19/24}\sum_{n\ge0}P_4(4n+3)q^n.
\]
These identities express residue-class subsequences of eta-power coefficients in terms of level-\(8\) eta quotients [2110.06768].

Several misconceptions are excluded by the current theory. First, level \(8\) does not imply novelty: at weight \(\tfrac12\), there is no primitive simple level-\(8\) object. Second, weak modularity is not the same as cusp holomorphy; the linear cusp-order constraints remain essential. Third, at level \(8\) the two Newman congruences are genuinely independent. Precisely because these issues are now explicit, level-\(8\) 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.

Source: https://www.emergentmind.com/topics/level-eight-eta-quotient