Atlas › Modular forms › Weight 2 › Level 457 › 457.2.a

Newspace 457.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(457))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
457.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} + q^{3} - q^{4} - 4q^{5} - 2q^{6} - q^{7} + O(q^{8})$	+	$-1$
457.2.a.b	15	degree 15	$15q - 10q^{2} - 12q^{3} + 16q^{4} - 9q^{5} + q^{6} - 15q^{7} + O(q^{8})$	+	$-1$
457.2.a.c	20	degree 20	$20q + 6q^{2} + 9q^{3} + 20q^{4} + 11q^{5} + 3q^{6} + 8q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
457.2.a.a	$x^{2} - 4x - 1$
457.2.a.b	$x^{15} + 46x^{14} + 753x^{13} + 3557x^{12} - 39225x^{11} - 549127x^{10} - 1558213x^{9} + 10229180x^{8} + 78781977x^{7} + 124652164x^{6} - 363414093x^{5} - 1324451877x^{4} - 496791642x^{3} + 2410780819x^{2} + 3061684520x + 1047726751$
457.2.a.c	$x^{20} - 33x^{19} + 141x^{18} + 6292x^{17} - 67885x^{16} - 375661x^{15} + 7801935x^{14} - 1633251x^{13} - 430160707x^{12} + 1189095187x^{11} + 12525762421x^{10} - 58048638457x^{9} - 176412964960x^{8} + 1273446526194x^{7} + 551230564192x^{6} - 13055160661257x^{5} + 11105864081886x^{4} + 46524964965279x^{3} - 73575290241975x^{2} + 19328340071067x - 364705139833$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(457))$	37
Old forms	0
Cusp forms $S_{2}(\Gamma_0(457))$	37
Eisenstein series $E_{2}(\Gamma_0(457))$	1
Modular forms $M_{2}(\Gamma_0(457))$	38

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.191 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, 457]: [0, 38] (certified at ell = 2147478649)
T = [(2, 1), (3, 3)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 5 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(457, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 457:2 1000

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