Atlas › Modular forms › Weight 2 › Level 763 › 763.2.a

Newspace 763.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(763))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
763.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} + q^{7} + O(q^{8})$	− −	$-1$	763a1 rank 1
763.2.a.b	10	degree 10	$10q - 5q^{2} - 2q^{3} + 9q^{4} - 10q^{5} - 3q^{6} + 10q^{7} + O(q^{8})$	− −	$-1$
763.2.a.c	13	degree 13	$13q - 3q^{2} - 6q^{3} + 13q^{4} - 14q^{5} - 3q^{6} - 13q^{7} + O(q^{8})$	+ +	$-1$
763.2.a.d	14	degree 14	$14q + 4q^{2} + 2q^{3} + 14q^{4} + 14q^{5} + 7q^{6} - 14q^{7} + O(q^{8})$	+ −	$+1$
763.2.a.e	17	degree 17	$17q + 7q^{2} + 2q^{3} + 23q^{4} + 12q^{5} - 5q^{6} + 17q^{7} + O(q^{8})$	− +	$+1$

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 $T$
763.2.a.a	$x + 2$
763.2.a.b	$x^{10} + 11x^{9} - 107x^{8} - 1194x^{7} + 3648x^{6} + 39417x^{5} - 55764x^{4} - 460568x^{3} + 348522x^{2} + 1738674x - 655065$
763.2.a.c	$x^{13} + 21x^{12} - 6x^{11} - 2401x^{10} - 7279x^{9} + 96439x^{8} + 317322x^{7} - 1931985x^{6} - 4267492x^{5} + 19408640x^{4} + 11844645x^{3} - 56190228x^{2} - 10840459x + 36543532$
763.2.a.d	$x^{14} - 10x^{13} - 244x^{12} + 2351x^{11} + 22102x^{10} - 195939x^{9} - 931947x^{8} + 6837506x^{7} + 18768456x^{6} - 89211968x^{5} - 160053241x^{4} + 349853539x^{3} + 334988628x^{2} - 65696228x - 65012356$
763.2.a.e	$x^{17} - 13x^{16} - 271x^{15} + 4012x^{14} + 24795x^{13} - 465410x^{12} - 887923x^{11} + 27166434x^{10} + 3120560x^{9} - 884143405x^{8} + 634590485x^{7} + 16451214844x^{6} - 17327754368x^{5} - 169655041208x^{4} + 177175878693x^{3} + 881317448234x^{2} - 610938044208x - 1835217971620$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(763))$	55
Old forms	16
Cusp forms $S_{2}(\Gamma_0(763))$	71
Eisenstein series $E_{2}(\Gamma_0(763))$	3
Modular forms $M_{2}(\Gamma_0(763))$	74

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(763)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(109))^{\oplus 2}$

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, 7, 109, 763]: [0, 1, 9, 74] (certified at ell = 2147482801)
T = [(2, 1), (3, 3)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 8 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(763, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 763:2 1000

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