Atlas › Modular forms › Weight 2 › Level 713 › 713.2.a

Newspace 713.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(713))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
713.2.a.a	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} + q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$	713a1 rank 1
713.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - 2q^{3} - q^{4} - 4q^{5} + 4q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
713.2.a.c	5	degree 5	$5q - 4q^{2} + 4q^{4} - 6q^{5} + 10q^{6} - 12q^{7} + O(q^{8})$	+ +	$-1$
713.2.a.d	5	degree 5	$5q + 4q^{4} + 5q^{5} - 14q^{6} - 9q^{7} + O(q^{8})$	+ +	$-1$
713.2.a.e	9	degree 9	$9q - 5q^{2} - 5q^{3} + 9q^{4} + q^{5} + q^{6} - 12q^{7} + O(q^{8})$	− −	$-1$
713.2.a.f	15	degree 15	$15q + q^{2} + 4q^{3} + 19q^{4} + q^{5} + 2q^{6} + 23q^{7} + O(q^{8})$	+ −	$+1$
713.2.a.g	18	degree 18	$18q + 5q^{2} + 2q^{3} + 25q^{4} + q^{5} + 4q^{6} + 23q^{7} + O(q^{8})$	− +	$+1$

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 $T$
713.2.a.a	$x - 4$
713.2.a.b	$x^{2} + 5x - 55$
713.2.a.c	$x^{5} + 4x^{4} - 119x^{3} - 477x^{2} + 1363x + 2521$
713.2.a.d	$x^{5} - 28x^{3} + 176x + 32$
713.2.a.e	$x^{9} + 20x^{8} + 53x^{7} - 941x^{6} - 4255x^{5} + 10963x^{4} + 55218x^{3} + 2932x^{2} - 52616x - 12048$
713.2.a.f	$x^{15} - 13x^{14} - 207x^{13} + 3242x^{12} + 11778x^{11} - 284982x^{10} - 30449x^{9} + 11431542x^{8} - 13937952x^{7} - 226250688x^{6} + 337235440x^{5} + 2394440880x^{4} - 2824262304x^{3} - 13042354112x^{2} + 7382578560x + 27053777664$
713.2.a.g	$x^{18} - 11x^{17} - 315x^{16} + 3351x^{15} + 39329x^{14} - 387049x^{13} - 2609631x^{12} + 21796294x^{11} + 102548464x^{10} - 633932929x^{9} - 2402253608x^{8} + 9277641200x^{7} + 30472997400x^{6} - 64249268704x^{5} - 183601428336x^{4} + 190476212576x^{3} + 454883377792x^{2} - 182408157696x - 331599840512$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(713))$	55
Old forms	8
Cusp forms $S_{2}(\Gamma_0(713))$	63
Eisenstein series $E_{2}(\Gamma_0(713))$	3
Modular forms $M_{2}(\Gamma_0(713))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(713)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.239 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, 23, 31, 713]: [0, 3, 3, 66] (certified at ell = 2147482921)
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(713, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 713:2 1000

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