Atlas › Modular forms › Weight 2 › Level 464 › 464.2.a

Newspace 464.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(464))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
464.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - 2q^{5} - 4q^{7} + O(q^{8})$	+	464e1 rank 0
464.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + q^{5} - 2q^{7} + O(q^{8})$	+	464b1 rank 1
464.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + 3q^{5} + 4q^{7} + O(q^{8})$	+	464d1 rank 0
464.2.a.d	1	$\mathbb{Q}$	$q + q^{3} - 3q^{5} - 2q^{7} + O(q^{8})$	+	464a1 rank 1
464.2.a.e	1	$\mathbb{Q}$	$q + q^{3} + q^{5} + 2q^{7} + O(q^{8})$	+	464c1 rank 0
464.2.a.f	1	$\mathbb{Q}$	$q + 3q^{3} - 3q^{5} + 2q^{7} + O(q^{8})$	+	464g1 rank 0
464.2.a.g	1	$\mathbb{Q}$	$q + 3q^{3} + 3q^{5} - 4q^{7} + O(q^{8})$	+	464f1 rank 0
464.2.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} - 2q^{5} + O(q^{8})$	−
464.2.a.i	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} - 2q^{5} + 8q^{7} + O(q^{8})$	−
464.2.a.j	3	degree 3	$3q - 2q^{3} + 4q^{5} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
464.2.a.a	$x - 38$
464.2.a.b	$x + 54$
464.2.a.c	$x - 70$
464.2.a.d	$x + 48$
464.2.a.e	$x - 32$
464.2.a.f	$x - 54$
464.2.a.g	$x - 18$
464.2.a.h	$x^{2} + 100x + 1618$
464.2.a.i	$x^{2} - 6962$
464.2.a.j	$x^{3} - 114x^{2} + 2766x - 18260$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(464))$	14
Old forms	41
Cusp forms $S_{2}(\Gamma_0(464))$	55
Eisenstein series $E_{2}(\Gamma_0(464))$	11
Modular forms $M_{2}(\Gamma_0(464))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(464)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(58))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(116))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(232))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.308 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 2, 4, 8, 16, 29, 58, 116, 232, 464]: [0, 1, 2, 3, 4, 3, 9, 18, 34, 64] (certified at ell = 2147483549)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 4 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(464, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 464:2 1000

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