---
title: Paramodular Newform Overview
url: https://www.emergentmind.com/topics/paramodular-newform
type: topic
---

# Paramodular Newform Overview

A paramodular newform is a degree-\(2\) Siegel modular form on the paramodular group \(K(N)\) that is new at level \(N\), equivalently a cuspidal Siegel eigenform whose attached automorphic representation of \(\mathrm{GSp}(4)\) has minimal local paramodular conductor \(N\); in the local representation-theoretic formulation, a paramodular newform at a finite place \(v\) is a vector fixed by \(K(\mathfrak p_v^{N_v})\) with \(N_v\) minimal among compact subgroups fixing a nonzero vector [1805.10873][2501.09109]. In weight \(2\), paramodular newforms are central to the Brumer–Kramer paramodularity conjecture, which relates typical abelian surfaces over \(\mathbb Q\) to nonlift paramodular newforms through equality of spinor and Hasse–Weil \(L\)-functions [1004.4699].

## 1. Ambient groups, level structure, and classical realization

The ambient algebraic group is
\[
\mathrm{GSp}(4)=\{\, g\in \mathrm{GL}(4)\mid {}^t g J g=\lambda(g)\,J \text{ for some } \lambda(g)\in \mathbb G_m\},
\]
with \(J=\begin{bmatrix}0&I_2\\-I_2&0\end{bmatrix}\). For a global level \(N\), the global paramodular subgroup \(K(N)\) is the subgroup of \(\mathrm{GSp}(4,\mathbb Q)\) consisting of matrices with the standard paramodular entry pattern
\[
K(N)=
\begin{bmatrix}
\mathbb Z & \mathbb Z & N^{-1}\mathbb Z & \mathbb Z\\
N\mathbb Z & \mathbb Z & \mathbb Z & \mathbb Z\\
N\mathbb Z & N\mathbb Z & \mathbb Z & N\mathbb Z\\
N\mathbb Z & \mathbb Z & \mathbb Z & \mathbb Z
\end{bmatrix}
\cap \mathrm{GSp}(4,\mathbb Q),
\]
and at a finite place \(v\) the local subgroup \(K(\mathfrak p_v^{N_v})\) is defined by the analogous \(\varpi_v^{\pm N_v}\)-integrality conditions in \(\mathrm{GSp}(4,L_v)\) [2501.09109]. In the rational setting, \(K(N)\) is also described as the natural modular group for \((1,N)\)-polarized abelian surfaces and as a stabilizer of a paramodular lattice [1004.4699].

A scalar-weight Siegel paramodular form is a holomorphic function \(F\) on the Siegel upper half-space \(\mathcal H\) satisfying
\[
F(\gamma\langle Z\rangle)=j(\gamma,Z)^{-k}F(Z),\qquad \gamma\in K(N),
\]
or, in slash notation, \(f|_k\gamma=f\). For arithmetic subgroups commensurable with \(\mathrm{Sp}_4(\mathbb Z)\), the resulting spaces \(M_k(K(N))\) and \(S_k(K(N))\) are the spaces of Siegel modular forms and cusp forms of degree \(2\) and level \(N\) [1805.10873][2501.09109].

The paramodular group is normalized by a Fricke involution. In the notation of the 2018 verification paper, \(K(N)\) is normalized by the paramodular Fricke involution \(U_N\), and the Fricke action gives a plus/minus decomposition
\[
M_k(K(N))=M_k(K(N))^{+}\oplus M_k(K(N))^{-}.
\]
This decomposition is fundamental in explicit constructions, especially for distinguishing lift and nonlift components and for organizing Atkin–Lehner eigenvalues [1805.10873].

## 2. Local newness, Hecke operators, and operator algebras

The local theory attaches to an irreducible admissible representation \((\pi,V)\) of \(\mathrm{GSp}(4,F)\) the spaces
\[
V(n)=\{v\in V:\pi(k)v=v\ \forall k\in K(\mathfrak p^n)\}.
\]
A representation is paramodular if \(V(n)\neq0\) for some \(n\), and its minimal such \(n\) is the paramodular level \(N_\pi\); the paramodular newform is a nonzero vector in \(V(N_\pi)\). For paramodular \(\pi\), one has \(\dim V(N_\pi)=1\), and for \(n\ge N_\pi\) the higher-level spaces are generated from the newvector by level-raising operators \(\eta,\theta,\theta'\) [2208.08939]. In the global cuspidal setting, Roberts–Schmidt newform theory implies that if \(f\) is new, then \(f\) is a Hecke eigenform at all \(p\) and all paramodular Atkin–Lehner involutions [1805.10873].

For primes \(p\nmid N\), the principal Hecke operators are
\[
T(p)=T\!\bigl(K(N)\operatorname{diag}(1,1,p,p)K(N)\bigr),
\qquad
T_1(p^2)=T\!\bigl(K(N)\operatorname{diag}(1,p,p^2,p)K(N)\bigr),
\]
and in weight \(2\) one also uses the integral combination \(B(p^2)\). If \(f\) is a common eigenform with eigenvalues \(a_p(f)\) and \(a_{1,p^2}(f)\), then the spinor Hecke polynomial is
\[
Q_p(f,T)=1-a_p(f)T+b_{p^2}(f)T^2-p^{2k-3}a_p(f)T^3+p^{4k-6}T^4,
\]
which in weight \(2\) becomes
\[
Q_p(f,T)=1-a_p(f)T+b_{p^2}(f)T^2-p\,a_p(f)T^3+p^2T^4.
\]
These polynomials are the Euler factors entering the spin \(L\)-function of the form [1805.10873].

The local structure at primes dividing the level is subtler. A paramodular newform at \(v\) is fixed by \(K(\mathfrak p_v^{N_v})\) with \(N_v\) minimal, and stable Klingen congruence subgroups provide an auxiliary filtration \(V_s(n)\) relating local paramodular vectors to three upper-block operators: two stable Klingen Hecke operators and one level-lowering operator. This yields a partition of paramodular representations into two classes, with category \(1\) characterized by the absence of unramified one-dimensional factors in the \(L\)-parameter decomposition; for generic \(\pi\), category \(1\) is equivalent to \(\mu_\pi\neq0\) [2208.08939].

A structural distinction also appears at the level of Hecke algebras. For squarefree \(N\), the Hecke algebra of the maximal discrete normal extension \(\Sigma_N^*\) is commutative and each local primary component is a polynomial ring in two algebraically independent generators, whereas the Hecke algebra of the non-maximal paramodular group \(\Sigma_N\) fails to be commutative if \(N>1\). In particular, at \(p\mid N\) the Atkin–Lehner double coset does not commute with one of the standard local Hecke operators, and zero-divisors appear in the non-maximal setting [1710.09156]. This clarifies why local newform theory is typically formulated either with the local \(p\)-adic paramodular subgroup or with the maximal extension rather than with the global non-maximal Hecke algebra.

## 3. Paramodularity, \(L\)-functions, and Galois representations

The arithmetic significance of paramodular newforms is encoded in the Brumer–Kramer conjecture. In its weight-\(2\) genus-\(2\) form, the conjecture asserts that if \(A/\mathbb Q\) is an abelian surface of conductor \(N\) with \(\operatorname{End}(A)=\mathbb Z\), then there exists a cuspidal paramodular newform
\[
f\in S_2(K(N))
\]
that is not a Gritsenko lift, has rational Hecke eigenvalues, is unique up to scaling, and satisfies
\[
L(A,s)=L(f,s,\mathrm{spin}).
\]
At a good prime \(p\nmid N\), the local factor of \(A\) has the shape
\[
L_p(A,T)=1-a_pT+b\,p^2T^2-p\,a_pT^3+p^2T^4,
\]
while the spinor polynomial of \(f\) has the shape
\[
Q_p(f,T)=1-a_p(f)T+b_{p^2}(f)T^2-p\,T^3+p^2T^4,
\]
and the conjectural identity becomes \(L_p(A,T)=Q_p(f,T)\) [1805.10873][1004.4699].

The associated Galois representations are likewise central. For a polarized abelian surface \(A/\mathbb Q\), the \(\ell\)-adic Tate module gives
\[
\rho_{A,\ell}: \operatorname{Gal}_{\mathbb Q,S}\to \mathrm{GSp}_4(\mathbb Z_\ell),
\]
and for a paramodular newform \(f\) of type \((G)\), one has a continuous semisimple representation
\[
\rho_{f,\ell}: \operatorname{Gal}_{\mathbb Q}\to \mathrm{GSp}_4(\overline{\mathbb Q}_\ell)
\]
with characteristic polynomial at good Frobenius equal to \(Q_p(f,T)\). Serre’s appendix on reduction of \(G\)-covariant bilinear forms ensures that residual representations retain the symplectic structure after semisimplification, which is decisive for comparison arguments in \(\mathrm{GSp}_4\) [1805.10873].

The comparison method used in rigorous cases generalizes the Faltings–Serre method from \(\mathrm{GL}_2\) to general \(G\subset \mathrm{GL}_n\). In the 2018 verification paper, a deterministic algorithm based only on Frobenius traces decides equivalence of two \(\ell\)-adic representations and produces a witness prime if they differ. The method combines residual comparison, deformation cocycles, class field theory, exact core-free subextensions, and local conductor control at primes dividing \(N\) [1805.10873].

The conjectural arithmetic class is the class of typical abelian surfaces, meaning \(A/\mathbb Q\) with \(\operatorname{End}(A_{\overline{\mathbb Q}})=\mathbb Z\). It is shown that \(A\) is typical if \(A\) is simple, semistable, and of nonsquare conductor, and in particular if the conductor is prime [1805.10873]. This typicality hypothesis excludes extra endomorphisms and isolates the genuinely degree-\(4\) case relevant to nonlift paramodular forms.

## 4. Construction mechanisms

Several independent constructions produce paramodular newforms or spaces in which paramodular newforms live. The basic additive construction is the Gritsenko lift from Jacobi cusp forms
\[
\mathrm{Grit}: J^{\mathrm{cusp}}_{k,N}\to S_k(K(N)),
\]
which is Hecke-equivariant and furnishes the lift subspace. In weight \(2\), nonlifts are defined by lying outside this image, and the prime-level construction paper completed the construction of all nonlift weight-two cusp paramodular Hecke eigenforms for prime levels \(N<600\), using Borcherds products and, at levels \(461\) and \(587^+\), trace-down from level \(2N\) [1805.04137].

Borcherds products provide the dominant explicit nonlift mechanism in low weight. In the antisymmetric setting, meromorphic weight-\(0\) Jacobi forms yield holomorphic antisymmetric paramodular Borcherds products whenever the Humbert multiplicities are nonnegative. This construction produces the antisymmetric nonlift in \(S_2(K(587))^{-}\), and further weight-\(2\) examples at levels \(713\) and \(893\), together with weight-\(3\) examples furnishing antisymmetric canonical differential forms on Siegel modular threefolds [1609.04146].

A distinct automorphic construction comes from Hilbert modular forms over real quadratic fields. Johnson–Leung and Roberts prove that if \(E/\mathbb Q\) is real quadratic and \(\pi_0\) is a cuspidal irreducible automorphic representation of \(\mathrm{GL}(2,\mathbb A_E)\) with trivial central character, infinity type \((2,2n+2)\), and not Galois invariant, then there exists a non-zero holomorphic Siegel paramodular newform \(F\) of weight \(k=n+2\) and explicitly determined paramodular level \(N\), with
\[
L_p(s+k-2,F)=L_p(s,\pi_0)
\]
for every prime \(p\) [1006.5105].

The 2025 theta-lift paper makes this construction explicit at the local level. For a real quadratic extension \(E/L\), it constructs local test data that produce a nonzero \(K(\mathfrak p_v^{N_v})\)-fixed vector in the local theta lift \(\Theta(\pi_v^+)\) at every finite place except when the local extension has wild ramification. The conductor exponent is determined directly from the \(\mathrm{GL}(2,E)\) conductor data:
\[
N_v=n_{1,v}+n_{2,v}\ \text{in the split case},\qquad
N_v=2n_v\ \text{in the inert unramified case},\qquad
N_v=n_v+2\ \text{in the tamely ramified case},
\]
and the paper states explicitly that no suitable Schwartz functions were found in the wildly ramified case [2501.09109].

Another source is the symmetric cube lift from non-CM elliptic curves. If \(E/\mathbb Q\) is a non-CM elliptic curve, then the symmetric cube transfer produces a holomorphic degree-\(2\) Siegel cusp form \(f\) of scalar weight \(3\), paramodular at level
\[
M=a(\operatorname{Sym}^3(\pi))=N\cdot \prod_{p\mid N,\ v_p(\Delta)\not\equiv 0\!\!\!\pmod 4} p^2,
\]
and satisfying
\[
L(s,f)=L(s,E,\operatorname{Sym}^3).
\]
The local conductor exponents are computed case by case from the reduction type of \(E\) and the local \(\mathrm{GL}_2\) representation \(\pi_p\) [1901.02115].

Finally, the term “paramodular newform” is not confined to cuspidal forms in every paper. A 2025 construction starts from a primitive Dirichlet character of conductor \(N\), produces a holomorphic Siegel Eisenstein series of paramodular level \(N^2\), computes its Fourier expansion, and states that the resulting function is a paramodular newform whose adelization generates an irreducible automorphic representation [2509.04395]. This shows that the newvector formalism extends beyond the cuspidal case, even though the arithmetic literature on abelian surfaces is mainly concerned with cuspidal nonlifts.

## 5. Computation, identification, and explicit extraction of Hecke data

The identification of a paramodular newform in explicit spaces depends on specialized computational frameworks. One such method is specialization to modular curves. For a suitable positive-definite symmetric matrix \(s\), the pullback
\[
\mathcal O_s^*:M(K(N),R)\to M(\Gamma_0(\det(s)N),R)
\]
doubles the weight and preserves cuspidality, converting a degree-\(2\) Siegel form into a one-variable \(q\)-series. In the 2018 verification paper, key cancellation identities reduce the cost of computing \(T(p)\) from \(O(p^3)\) sums to \(O(p^2)\), making the Hecke eigenvalues \(a_p(f)\) and \(b_{p^2}(f)\) accessible by coefficient comparison [1805.10873].

At primes with \(p^2\mid N\), classical paramodular Hecke operators do not admit simple upper-block formulas. The stable Klingen framework replaces them by \(T_{0,1}^s(p)\), \(T_{1,0}^s(p)\), and a level-lowering operator \(\sigma_p\). For a newform \(F\in S_k(K(N))\), these operators yield explicit criteria for \(\mu_p=0\) or \(\mu_p\neq0\), formulas recovering \(\lambda_p\) and \(\mu_p\) from Fourier coefficients, and a rational generating function for radial coefficients whose denominator is \(L_p(s,F)^{-1}\). In particular, when \(v_p(N)\ge2\), the paper shows
\[
L_p(s,F)=\frac{1}{1-p^{k-3}\lambda_p p^{-s}+p^{2k-5}(\mu_p+p^2)p^{-2s}}
\]
in the stable Klingen normalization [2208.08939].

A different computational bridge is provided by algebraic modular forms and quinary lattices. For squarefree levels, the exact relationship between algebraic modular forms on a definite quaternionic unitary group and genera of positive-definite quinary lattices identifies the Hecke operator \(T(p)\) with the \(p\)-neighbor operator and \(T_1(p^2)\) with the \(p^2\)-neighbor operator. The resulting adjacency matrices can be diagonalized to produce Hecke eigenvalues for paramodular forms of general type, and the Atkin–Lehner signs are isolated by explicit sign characters \(\theta_d\) [2112.03797].

In the low-weight, low-level computations of weight \(2\), Jacobi restriction remains a principal tool. It organizes a paramodular form through finitely many Fourier–Jacobi coefficients, constructs finite-dimensional superspaces \(J^{(d)}\) of truncated data, and, together with reduction modulo a prime and Atkin–Lehner “infilling,” certifies dimensions and nonlift status. This method underlies the determination of \(S_2(K(N))\) for squarefree \(N<300\) and of nonlift prime-level forms below \(600\) [1612.00925][1805.04137].

## 6. Explicit cases, congruences, and current extensions

The strongest complete verifications currently available are the prime-conductor cases \(N=277\), \(353\), and \(587\). For \(N=277\), there is a unique nonlift \(f_{277}\in S_2(K(277),\mathbb Z)^{+}\), constructed as a rational function of ten Gritsenko lifts of weight-\(2\) theta blocks; the paper computes Hecke data up to \(p\le43\), compares traces of the associated mod-\(2\) Galois representations at a finite witness set, and proves
\[
L(A,s)=L(f_{277},s,\mathrm{spin})
\]
for the Jacobian \(A=\operatorname{Jac}(X)\) of conductor \(277\). The same strategy proves the analogous statements for \(N=353\) and for the antisymmetric Borcherds product \(f_{587}\in S_2(K(587),\mathbb Z)^{-}\) [1805.10873].

At composite squarefree level below \(300\), the structure is more rigid. The space \(S_2(K(N))\) equals the additive Gritsenko lift space for all composite squarefree \(N<300\) except \(N=249\) and \(N=295\), and in each of these two exceptional cases there is exactly one additional nonlift newform line. For those levels, the spin \(p\)-Euler factors of the nonlift newform agree with the Hasse–Weil factors of the associated abelian surface at the first two good primes \(p\nmid N\) [1612.00925]. A common simplification is therefore incorrect: a paramodular space need not contain nonlifts, and at many levels it is entirely accounted for by the Gritsenko lift space.

The role of congruences has also expanded. In the quinary-lattice framework, examples of Harder-type and Buzzard–Golyshev-type congruences connect paramodular eigenvalues to elliptic modular forms at levels \(61\), \(89\), \(19\), \(42\), \(13\), and \(5\) [2112.03797]. In a more geometric direction, a 2024 paper proves residual paramodularity for a Calabi–Yau threefold: the unique nonlift Hecke eigenform
\[
F_{79}\in S_3(K(79))
\]
is shown to satisfy congruences modulo a prime above \(5\) with a Johnson–Leung–Roberts lift \(\mathrm{JR}(h_{79})\), and the semisimplified mod-\(5\) Galois representation on \(H^3\) of the threefold is identified with the residual representation attached to \(F_{79}\) [2412.14289].

These examples suggest two complementary themes. First, the classical weight-\(2\) nonlift case tied to abelian surfaces remains the core arithmetic application. Second, the same local-newvector and Hecke-theoretic formalism now supports weight-\(3\) lifts, residual modularity for four-dimensional motives, and even paramodular Eisenstein newforms. The modern theory of paramodular newforms therefore sits simultaneously in explicit Siegel modular form computation, local representation theory on \(\mathrm{GSp}(4)\), and the arithmetic of degree-\(4\) \(L\)-functions.

Source: https://www.emergentmind.com/topics/paramodular-newform