Atlas › Modular forms › Weight 2 › Level 472 › 472.2.a

Newspace 472.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(472))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
472.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - q^{5} + 3q^{7} + O(q^{8})$	+	472a1 rank 1
472.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + q^{7} + O(q^{8})$	−	472e1 rank 1
472.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + q^{7} + O(q^{8})$	−	472b1 rank 0
472.2.a.d	1	$\mathbb{Q}$	$q + 2q^{3} + 2q^{5} + q^{7} + O(q^{8})$	−	472c1 rank 0
472.2.a.e	1	$\mathbb{Q}$	$q + 3q^{3} - 3q^{5} + 3q^{7} + O(q^{8})$	+	472d1 rank 0
472.2.a.f	4	degree 4	$4q - q^{3} - 3q^{5} - 9q^{7} + O(q^{8})$	+
472.2.a.g	6	degree 6	$6q - q^{3} + 7q^{5} - 4q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
472.2.a.a	$x + 38$
472.2.a.b	$x + 72$
472.2.a.c	$x - 48$
472.2.a.d	$x + 3$
472.2.a.e	$x + 36$
472.2.a.f	$x^{4} + 91x^{3} + 1918x^{2} - 15927x - 409712$
472.2.a.g	$x^{6} - 194x^{5} + 8963x^{4} + 65099x^{3} - 11378297x^{2} + 220947332x - 1277304748$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(472))$	15
Old forms	42
Cusp forms $S_{2}(\Gamma_0(472))$	57
Eisenstein series $E_{2}(\Gamma_0(472))$	7
Modular forms $M_{2}(\Gamma_0(472))$	64

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(472)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(59))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(118))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(236))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.359 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, 59, 118, 236, 472]: [0, 1, 2, 3, 6, 17, 33, 64] (certified at ell = 2147482481)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 15 = 1 x 15 (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(472, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 472:2 1000

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