Atlas › Modular forms › Weight 2 › Level 476 › 476.2.a

Newspace 476.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(476))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
476.2.a.a	2	$\mathbb{Q}(\sqrt{13})$	$2q - 3q^{3} - 3q^{5} + 2q^{7} + O(q^{8})$	− +
476.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{3} - q^{5} - 2q^{7} + O(q^{8})$	+ −
476.2.a.c	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{3} + q^{5} - 2q^{7} + O(q^{8})$	+ +
476.2.a.d	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{3} - q^{5} + 2q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
476.2.a.a	$x^{2} + 12x + 23$
476.2.a.b	$x^{2} + 4x - 1$
476.2.a.c	$x^{2} - 2x - 51$
476.2.a.d	$x^{2} + 2x - 51$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(476))$	8
Old forms	59
Cusp forms $S_{2}(\Gamma_0(476))$	67
Eisenstein series $E_{2}(\Gamma_0(476))$	11
Modular forms $M_{2}(\Gamma_0(476))$	78

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(476)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(68))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(119))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(238))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.251 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, 7, 14, 17, 28, 34, 68, 119, 238, 476]: [0, 1, 2, 1, 4, 2, 7, 6, 12, 14, 40, 78] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(476, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 476:2 1000

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