Atlas › Modular forms › Weight 2 › Level 707 › 707.2.a

Newspace 707.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(707))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
707.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} + 4q^{6} - q^{7} + O(q^{8})$	+ −	$+1$	707a1 rank 2
707.2.a.b	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} + 2q^{3} + 3q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
707.2.a.c	4	degree 4	$4q + 4q^{3} - 8q^{4} + q^{5} + 4q^{7} + O(q^{8})$	− +	$+1$
707.2.a.d	8	degree 8	$8q - 3q^{2} - 3q^{3} + 3q^{4} - 2q^{5} + 2q^{6} - 8q^{7} + O(q^{8})$	+ +	$-1$
707.2.a.e	10	degree 10	$10q - 2q^{2} - 9q^{3} + 6q^{4} - 7q^{5} - 5q^{6} + 10q^{7} + O(q^{8})$	− −	$-1$
707.2.a.f	11	degree 11	$11q + 2q^{2} + 3q^{3} + 22q^{4} + 4q^{5} - q^{6} + 11q^{7} + O(q^{8})$	− +	$+1$
707.2.a.g	15	degree 15	$15q + 5q^{2} + q^{3} + 19q^{4} + 10q^{5} + q^{6} - 15q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
707.2.a.a	$x + 4$
707.2.a.b	$x^{2} - x - 3$
707.2.a.c	$x^{4} - 4x^{3} - x^{2} + 9x + 4$
707.2.a.d	$x^{8} + 6x^{7} - 8x^{6} - 72x^{5} - 2x^{4} + 232x^{3} + 75x^{2} - 145x - 4$
707.2.a.e	$x^{10} + 11x^{9} + 33x^{8} - 18x^{7} - 188x^{6} - 76x^{5} + 301x^{4} + 92x^{3} - 128x^{2} - 9x + 4$
707.2.a.f	$x^{11} - 5x^{10} - 29x^{9} + 150x^{8} + 246x^{7} - 1312x^{6} - 937x^{5} + 3816x^{4} + 2376x^{3} - 2233x^{2} - 778x + 74$
707.2.a.g	$x^{15} - 6x^{14} - 39x^{13} + 248x^{12} + 514x^{11} - 3423x^{10} - 3308x^{9} + 19851x^{8} + 12193x^{7} - 47809x^{6} - 18726x^{5} + 48209x^{4} + 10198x^{3} - 17201x^{2} - 786x + 146$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(707))$	51
Old forms	16
Cusp forms $S_{2}(\Gamma_0(707))$	67
Eisenstein series $E_{2}(\Gamma_0(707))$	3
Modular forms $M_{2}(\Gamma_0(707))$	70

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(707)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(101))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.319 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, 101, 707]: [0, 1, 9, 70] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 51 = 1 x 51 (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(707, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 707:2 1000

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