Atlas › Modular forms › Weight 2 › Level 771 › 771.2.a

Newspace 771.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(771))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
771.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} + 2q^{3} - q^{4} - 2q^{5} + q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
771.2.a.b	6	degree 6	$6q - 6q^{2} + 6q^{3} + 6q^{4} - 9q^{5} - 6q^{6} + O(q^{8})$	− −	$-1$
771.2.a.c	8	degree 8	$8q + q^{2} - 8q^{3} + 5q^{4} - 7q^{5} - q^{6} - 6q^{7} + O(q^{8})$	+ +	$-1$
771.2.a.d	13	degree 13	$13q + 5q^{2} + 13q^{3} + 13q^{4} + 11q^{5} + 5q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$
771.2.a.e	14	degree 14	$14q - 2q^{2} - 14q^{3} + 20q^{4} + 9q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
771.2.a.a	$x^{2} + x - 1$
771.2.a.b	$x^{6} + 15x^{5} + 81x^{4} + 176x^{3} + 63x^{2} - 246x - 199$
771.2.a.c	$x^{8} + 6x^{7} + x^{6} - 30x^{5} + 4x^{4} + 45x^{3} - 38x^{2} + 11x - 1$
771.2.a.d	$x^{13} - 16x^{12} + 68x^{11} + 169x^{10} - 1989x^{9} + 3502x^{8} + 9306x^{7} - 39222x^{6} + 32856x^{5} + 38237x^{4} - 78164x^{3} + 35313x^{2} + 4538x - 4658$
771.2.a.e	$x^{14} - 7x^{13} - 60x^{12} + 455x^{11} + 1152x^{10} - 10655x^{9} - 6824x^{8} + 112886x^{7} - 22144x^{6} - 556529x^{5} + 314495x^{4} + 1258461x^{3} - 940893x^{2} - 1058036x + 901802$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(771))$	43
Old forms	42
Cusp forms $S_{2}(\Gamma_0(771))$	85
Eisenstein series $E_{2}(\Gamma_0(771))$	3
Modular forms $M_{2}(\Gamma_0(771))$	88

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(771)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(257))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 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, 3, 257, 771]: [0, 1, 22, 88] (certified at ell = 2147483137)
T = [(2, 1), (5, 1)]; h of degree 43 = 1 x 43 (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(771, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 771:2 1000

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