Atlas › Modular forms › Weight 2 › Level 712 › 712.2.a

Newspace 712.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(712))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
712.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + 2q^{5} - 4q^{7} + O(q^{8})$	−	712a1 rank 0
712.2.a.b	3	degree 3	$3q - q^{3} - 3q^{5} - 2q^{7} + O(q^{8})$	+
712.2.a.c	5	degree 5	$5q + 3q^{3} + q^{5} + 2q^{7} + O(q^{8})$	+
712.2.a.d	6	degree 6	$6q - 5q^{3} - 3q^{5} - 2q^{7} + O(q^{8})$	−
712.2.a.e	7	degree 7	$7q + 3q^{3} + 3q^{5} + 6q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
712.2.a.a	$x - 4$
712.2.a.b	$x^{3} + 10x^{2} + 30x + 26$
712.2.a.c	$x^{5} - 6x^{4} - 128x^{3} + 418x^{2} + 1808x + 32$
712.2.a.d	$x^{6} + 14x^{5} - 46x^{4} - 850x^{3} + 760x^{2} + 9526x + 9668$
712.2.a.e	$x^{7} - 12x^{6} - 230x^{5} + 3254x^{4} + 6040x^{3} - 205942x^{2} + 826728x - 977936$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(712))$	22
Old forms	65
Cusp forms $S_{2}(\Gamma_0(712))$	87
Eisenstein series $E_{2}(\Gamma_0(712))$	7
Modular forms $M_{2}(\Gamma_0(712))$	94

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(712)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(89))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(178))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(356))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.249 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, 89, 178, 356, 712]: [0, 1, 2, 3, 8, 24, 48, 94] (certified at ell = 2147482921)
T = [(3, 1), (5, 3)]; h of degree 22 = 1 x 22 (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(712, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 712:2 1000

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