Atlas › Modular forms › Weight 2 › Level 456 › 456.2.a

Newspace 456.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(456))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
456.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + q^{5} - 3q^{7} + O(q^{8})$	+ −	456d1 rank 1
456.2.a.b	1	$\mathbb{Q}$	$q - q^{3} + 4q^{5} + 4q^{7} + O(q^{8})$	+ −	456a1 rank 0
456.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - 3q^{5} - 3q^{7} + O(q^{8})$	− −	456c1 rank 1
456.2.a.d	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} + O(q^{8})$	− +	456b1 rank 0
456.2.a.e	2	$\mathbb{Q}(\sqrt{41})$	$2q - 2q^{3} - q^{5} - 3q^{7} + O(q^{8})$	+ −
456.2.a.f	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{3} + q^{5} + q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{23} + 11T_{29}$ 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 $T$
456.2.a.a	$x + 55$
456.2.a.b	$x - 6$
456.2.a.c	$x + 159$
456.2.a.d	$x - 48$
456.2.a.e	$x^{2} - 173x + 6652$
456.2.a.f	$x^{2} - 91x + 2066$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(456))$	8
Old forms	65
Cusp forms $S_{2}(\Gamma_0(456))$	73
Eisenstein series $E_{2}(\Gamma_0(456))$	15
Modular forms $M_{2}(\Gamma_0(456))$	88

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(456)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 8}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(24))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(57))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(76))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(114))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(152))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(228))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.362 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, 3, 4, 6, 8, 12, 19, 24, 38, 57, 76, 114, 152, 228, 456]: [0, 1, 1, 2, 3, 3, 5, 2, 8, 7, 8, 13, 24, 24, 46, 88] (certified at ell = 2147483647)
T = [(5, 1), (7, 3), (13, 5), (17, 7), (23, 9), (29, 11)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 3 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(456, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 456:2 1000

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