Primitive Eta-Products in Modular Forms

• The paper shows that in weight 1/2, among fourteen primitive holomorphic eta quotients, only η itself qualifies as an eta-product when all exponents are nonnegative.
• Later research distinguishes level-primitive, exponent-primitive, and Hecke-primitive forms, emphasizing that primitivity is context-dependent and essential for modular classifications.
• Applications extend to theta-series representations, multiplicative completions, moonshine phenomena, and connections with K3 geometries and supersymmetric partition functions.

Primitive eta-products are Dedekind-eta products viewed through a primitivity condition that excludes trivial inflation by a common scaling, although the precise meaning of “primitive” varies across the literature. In the modular-form setting of holomorphic eta quotients, one begins with

$$\eta(z)=q^{1/24}\prod_{n\ge 1}(1-q^n),\qquad q=e^{2\pi i z},$$

and an eta quotient

$$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$

of level

$N=\operatorname{lcm}\{d\in\mathbb N:r_d\neq 0\}$

and weight

$k=\frac12\sum_{d\mid N} r_d.$

If all exponents satisfy $r_d\ge 0$, then $f$ is an eta-product; otherwise it is an eta quotient (Bhattacharya, 2016). For weight $1/2$, the subject admits a complete classification: every holomorphic eta quotient is an integral rescaling of one of fourteen primitive holomorphic eta quotients, and among those fourteen only $\eta$ itself is an eta-product (Bhattacharya, 2016). Beyond that classical classification, later work uses several non-equivalent notions of primitivity, including level-primitive, exponent-primitive, Hecke-primitive, and primitive frame-shapes, so the term has to be read contextually (Aydin et al., 2023, He et al., 2013).

1. Basic definitions and competing notions of primitivity

In the strict sense used for holomorphic eta quotients of weight $1/2$, an eta quotient $f$ is primitive if there do not exist an eta quotient $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$0 and an integer $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$1 such that

$$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$2

This operation is called an integral rescaling. If $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$3 has level $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$4, then $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$5 has level $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$6; equivalently, rescaling multiplies the arguments of all $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$7 by $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$8 (Bhattacharya, 2016). In this framework, a primitive eta-product is simply a primitive holomorphic eta quotient whose exponents are all nonnegative (Bhattacharya, 2016).

A substantial source of ambiguity is that later papers deliberately separate several notions that the older literature often conflated. One study of half-integral Hecke eigenforms states explicitly that it does not impose or classify “primitive” in the sense of level or newness, and then distinguishes level-primitive, exponent-primitive, and Hecke-primitive forms (Aydin et al., 2023). In work on multiplicative eta-products, primitive frame-shapes are described in terms of the support of the product, with a further possible exclusion of perfect powers (He et al., 2013). By contrast, a paper on weight $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$9 eta-quotients and binary quadratic forms uses “primitive” for the underlying quadratic forms, not for eta-products themselves (Berkovich et al., 2013).

Setting	Primitive means	Reference
Holomorphic eta quotients	not of the form $N=\operatorname{lcm}\{d\in\mathbb N:r_d\neq 0\}$0 with $N=\operatorname{lcm}\{d\in\mathbb N:r_d\neq 0\}$1	(Bhattacharya, 2016)
Level-primitive	$N=\operatorname{lcm}\{d\in\mathbb N:r_d\neq 0\}$2	(Aydin et al., 2023)
Exponent-primitive	$N=\operatorname{lcm}\{d\in\mathbb N:r_d\neq 0\}$3	(Aydin et al., 2023)
Hecke-primitive	Shimura lift is a newform	(Aydin et al., 2023)
Primitive frame-shape	support has gcd $N=\operatorname{lcm}\{d\in\mathbb N:r_d\neq 0\}$4; sometimes also excludes power forms	(He et al., 2013)
Primitive binary quadratic form	$N=\operatorname{lcm}\{d\in\mathbb N:r_d\neq 0\}$5 for the quadratic form	(Berkovich et al., 2013)

This terminological dispersion is not merely linguistic. It affects classification statements, because a form may be primitive under one criterion and imprimitive under another. A rescaling-primitive eta-product can fail to be Hecke-primitive, and a product that is level-imprimitive may still be important as a Hecke eigenform or as a multiplicative object.

2. Modularity, levels, and cusp-order control

The natural modular group attached to an eta quotient of level $N=\operatorname{lcm}\{d\in\mathbb N:r_d\neq 0\}$6 is

$N=\operatorname{lcm}\{d\in\mathbb N:r_d\neq 0\}$7

the largest congruence subgroup on which the eta quotient transforms as a modular form with the inherited multiplier (Bhattacharya, 2016). Holomorphy is determined by the behavior at the cusps.

For $N=\operatorname{lcm}\{d\in\mathbb N:r_d\neq 0\}$8 with $N=\operatorname{lcm}\{d\in\mathbb N:r_d\neq 0\}$9, the order of $k=\frac12\sum_{d\mid N} r_d.$0 at $k=\frac12\sum_{d\mid N} r_d.$1 is

$k=\frac12\sum_{d\mid N} r_d.$2

and hence for

$k=\frac12\sum_{d\mid N} r_d.$3

one has

$k=\frac12\sum_{d\mid N} r_d.$4

Holomorphy at all cusps is equivalent to nonnegative cusp orders (Bhattacharya, 2016).

A useful packaging of these orders is the order matrix $k=\frac12\sum_{d\mid N} r_d.$5, defined by

$k=\frac12\sum_{d\mid N} r_d.$6

If $k=\frac12\sum_{d\mid N} r_d.$7, then the vector of cusp orders is

$k=\frac12\sum_{d\mid N} r_d.$8

so holomorphy is equivalent to $k=\frac12\sum_{d\mid N} r_d.$9 entrywise (Bhattacharya, 2016). The matrix $r_d\ge 0$0 is multiplicative in $r_d\ge 0$1,

$r_d\ge 0$2

and its prime-power blocks are explicitly invertible; in particular, an eta quotient is uniquely determined by its cusp orders (Bhattacharya, 2016).

A parallel formulation in the half-integral Hecke-eigenform literature uses Newman–Ligozat congruences. For

$r_d\ge 0$3

standard conditions include

$r_d\ge 0$4

together with nonnegative cusp orders, to place $r_d\ge 0$5 in $r_d\ge 0$6 or $r_d\ge 0$7 (Aydin et al., 2023). These criteria do not define primitivity, but they delimit the ambient modular spaces within which primitive objects are sought.

3. The weight $r_d\ge 0$8 classification and its consequences

The decisive structural result is the theorem identified as Zagier’s conjecture, proved by Mersmann: each holomorphic eta quotient of weight $r_d\ge 0$9 is a rescaling by a positive integer of one of fourteen primitive holomorphic eta quotients (Bhattacharya, 2016). Their levels lie among

$f$0

and every member of the list has total exponent sum $f$1, hence weight $f$2 (Bhattacharya, 2016).

For the specific topic of primitive eta-products, the key point is sharper: among Zagier’s fourteen primitive holomorphic eta quotients of weight $f$3, only $f$4 itself is an eta-product; all others are genuine quotients with some negative exponents (Bhattacharya, 2016). Thus, in weight $f$5, primitive eta-products collapse to a single primitive model, namely $f$6, while nonprimitive eta-products arise from integral rescalings.

The proof strategy proceeds in five steps. First, cusp orders are encoded by the order matrix $f$7 and its symmetrized form $f$8. Second, one constructs holomorphy-preserving homomorphisms $f$9 on eta quotients. Third, these maps reduce the classification to 3-smooth levels. Fourth, bounds at $1/2$0-power levels imply that if a primitive holomorphic eta quotient of weight $1/2$1 has level $1/2$2, then necessarily $1/2$3 and $1/2$4. Fifth, a direct linear-algebra check on $1/2$5 leaves exactly the fourteen primitive forms (Bhattacharya, 2016).

Two consequences are especially relevant. The first is computational: by applying the Jacobi triple product identity, the paper obtains theta-series representations

$1/2$6

for the primitive list, with $1/2$7, and then any holomorphic weight $1/2$8 eta quotient is handled by rescaling $1/2$9 (Bhattacharya, 2016). The second is factorization-theoretic: since $\eta$0 is the smallest possible weight of any holomorphic eta quotient, no holomorphic eta quotient of weight $\eta$1 factors nontrivially. Hence simplicity and primitivity coincide in this weight (Bhattacharya, 2016).

The same paper also derives an extension principle for levels. If there exists a simple, respectively irreducible, holomorphic eta quotient of odd level $\eta$2, then there are at least two simple, respectively irreducible, holomorphic eta quotients of level $\eta$3 and at least three of level $\eta$4; if $\eta$5, there are also four such eta quotients of levels $\eta$6 and $\eta$7 (Bhattacharya, 2016). This does not create new primitive eta-products in weight $\eta$8, but it shows how primitive low-weight building blocks propagate through higher levels.

4. Half-integral Hecke eigenforms and refined notions of primitiveness

A later classification problem concerns Dedekind-eta products of half-integral weight that are Hecke eigenforms up to weight $\eta$9. In that setting, the paper’s emphasis is not on a single primitivity notion, and it explicitly states that it does not impose or classify “primitive” in the sense of level or newness (Aydin et al., 2023). Instead, it proposes a three-way distinction.

Level-primitive means that the support of the eta-product has gcd $1/2$0, so the product cannot be obtained from a strictly smaller level by a common scaling of all arguments. Exponent-primitive means that the exponent vector has gcd $1/2$1, so the product is not a nontrivial power of another eta-product with the same arguments. Hecke-primitive, or newform-primitive, means that the Shimura lift is a newform of the appropriate integral weight and level (Aydin et al., 2023). This vocabulary is useful precisely because the three conditions need not coincide.

The modular setting is the half-integral space $1/2$2, together with Kohnen’s plus space and the Hecke operators $1/2$3 for primes $1/2$4 (Aydin et al., 2023). The paper uses Purkait’s generating set for the relevant Hecke algebra, a Sturm bound, and the Shimura correspondence

$1/2$5

to certify eigenform status (Aydin et al., 2023).

Its Theorem 2.1 and Table 1 give a complete list of Dedekind-eta products that are Hecke eigenforms of half-integral weight for all weights up to $1/2$6, at various levels and characters (Aydin et al., 2023). Even in weight $1/2$7, the listed examples $1/2$8 and $1/2$9 are not level-primitive, since their supports have gcd $f$0 and $f$1, respectively (Aydin et al., 2023). The broader implication is that primitivity and Hecke-theoretic significance are logically distinct: eta-products that are imprimitive by scaling can still be distinguished Hecke eigenforms.

5. Weight $f$2, binary quadratic forms, and multiplicative completion

In the study of certain weight $f$3 eta-quotients, the central mechanism is an identity that expresses eta-products in terms of binary quadratic forms. For positive integers $f$4 with $f$5,

$f$6

where $f$7 is the theta series of the quadratic form $f$8 and $f$9 (Berkovich et al., 2013). Since $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$00, the right-hand side is a weight $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$01 eta-product up to an overall $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$02-power.

Here the word “primitive” applies explicitly to binary quadratic forms, with $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$03 primitive when $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$04 (Berkovich et al., 2013). The paper does not define “primitive eta-product” as a separate notion. Instead, the eta-products are realized via differences of theta series of primitive forms in the same genus, and then completed to Hecke eigenforms by adding suitable linear combinations of theta series.

This procedure is called multiplicative completion. A completion $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$05 of an eta-quotient $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$06 is a linear combination with another $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$07-series $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$08, constructed from theta series, such that $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$09 is a Hecke eigenform with multiplicative coefficients (Berkovich et al., 2013). The supports of $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$10 and $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$11 are taken to be congruentially disjoint, so that the coefficients of $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$12 can be extracted from the multiplicative completion. Explicit coefficient formulas are then obtained for eta-quotients at levels $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$13, $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$14, $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$15, $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$16, $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$17, and $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$18 (Berkovich et al., 2013).

This framework shows that eta-products can be primitive in a representation-theoretic or arithmetic sense without the literature ever fixing a single standalone definition of “primitive eta-product.” A plausible implication is that, in weight $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$19, the more robust structure is often not primitivity by rescaling, but the passage from eta-products to Hecke eigenforms through genus theory and theta-series identities.

6. Multiplicative frame-shapes, geometry, moonshine, and physics

A different but influential usage arises in the classification of multiplicative eta-products associated with partitions of $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$20. In frame-shape notation,

$$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$21

and, more generally,

$$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$22

has weight $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$23 (He et al., 2013). The Dummit–Kisilevsky–McKay classification identifies exactly $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$24 eta-products among the $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$25 partitions of $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$26 whose Fourier coefficients are multiplicative (He et al., 2013).

In that setting, primitive by scaling means

$$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$27

so that no common scaling can be factored out. A second, stricter convention also excludes power forms by requiring $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$28 (He et al., 2013). Primitive examples include

$$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$29

whereas forms such as $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$30, $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$31, $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$32, and $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$33 are imprimitive by scaling (He et al., 2013). The paper further notes that $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$34 of the $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$35 multiplicative eta-products correspond to conjugacy classes of $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$36, with $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$37 exceptions (He et al., 2013).

The same work matches several primitive multiplicative eta-products to elliptic K3 surfaces admitting Nikulin automorphisms and to CHL-type BPS-state generating functions. In particular, four key primitive shapes,

$$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$38

are identified with the congruence subgroups $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$39, $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$40, $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$41, and $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$42, and with the Nikulin types $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$43, $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$44, $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$45, and $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$46, respectively (He et al., 2013). Here primitiveness is inseparable from the frame-shape formalism and from the presence of a cycle length $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$47 in the support.

Recent work in $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$48-dimensional supersymmetric Chern–Simons theory provides yet another context. That paper does not define “primitive” explicitly, but it records a natural criterion, standard in the literature: an eta-product is primitive if the set of arguments has gcd $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$49, so the form cannot be obtained by a uniform scaling (Okazaki et al., 2024). Under that criterion,

$$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$50

and

$$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$51

are described as primitive, while the adjoint-only family

$$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$52

is not primitive for any $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$53, since its arguments have gcd $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$54 (Okazaki et al., 2024). The paper summarizes the resulting pattern succinctly: eta-products with odd moduli mixed with $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$55 tend to be primitive, whereas many even-modulus families anchored at $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$56 are not (Okazaki et al., 2024).

Across these literatures, primitive eta-products are therefore best understood not as a single invariant class but as a family of closely related notions. In the strict rescaling sense, the weight $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$57 theory is completely rigid and leaves only $$f(z)=\prod_{d\mid N}\eta(dz)^{r_d},\qquad r_d\in\mathbb Z,$$58 as the primitive eta-product (Bhattacharya, 2016). In Hecke theory, binary quadratic forms, multiplicative moonshine, K3 geometry, and supersymmetric partition functions, the same term is reused for support-gcd, exponent-gcd, or newform-type conditions. The common structural theme is that primitivity singles out eta-products not obtained by an obvious inflation, and the sharpest classifications arise when that exclusion is combined with explicit modularity and cusp-order control.