Atlas › Modular forms › Weight 2 › Level 913 › 913.2.a

Newspace 913.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(913))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
913.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - q^{7} + O(q^{8})$	+ −	$+1$	913a1 rank 0
913.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + 3q^{3} + 2q^{4} + 3q^{5} + 6q^{6} - 4q^{7} + O(q^{8})$	+ −	$+1$	913b1 rank 0
913.2.a.c	5	degree 5	$5q + q^{2} + q^{3} + q^{4} + 3q^{5} + 2q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$
913.2.a.d	12	degree 12	$12q + 2q^{2} + 4q^{3} + 20q^{4} - q^{5} - 9q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$
913.2.a.e	15	degree 15	$15q - 7q^{2} - 7q^{3} + 15q^{4} - 4q^{5} + 3q^{6} - 17q^{7} + O(q^{8})$	− −	$-1$
913.2.a.f	16	degree 16	$16q - 5q^{2} - 8q^{3} + 13q^{4} - 7q^{5} - 5q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
913.2.a.g	17	degree 17	$17q + 9q^{2} + 5q^{3} + 17q^{4} + 2q^{5} + 3q^{6} + 13q^{7} + O(q^{8})$	− +	$+1$

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 $T$
913.2.a.a	$x + 1$
913.2.a.b	$x - 5$
913.2.a.c	$x^{5} - 2x^{4} - 13x^{3} + 31x^{2} + 7x - 35$
913.2.a.d	$x^{12} - 6x^{11} - 25x^{10} + 152x^{9} + 279x^{8} - 1230x^{7} - 2061x^{6} + 3162x^{5} + 7414x^{4} + 1938x^{3} - 3897x^{2} - 2848x - 535$
913.2.a.e	$x^{15} + 14x^{14} + 43x^{13} - 203x^{12} - 1298x^{11} - 673x^{10} + 7363x^{9} + 9917x^{8} - 16343x^{7} - 28829x^{6} + 15729x^{5} + 31632x^{4} - 5587x^{3} - 11875x^{2} - 241x + 681$
913.2.a.f	$x^{16} + 13x^{15} + 37x^{14} - 170x^{13} - 975x^{12} + 21x^{11} + 7204x^{10} + 8056x^{9} - 19536x^{8} - 37810x^{7} + 9008x^{6} + 53291x^{5} + 22733x^{4} - 12618x^{3} - 7232x^{2} + 804x + 369$
913.2.a.g	$x^{17} - 14x^{16} + 36x^{15} + 341x^{14} - 2099x^{13} + 962x^{12} + 19245x^{11} - 45968x^{10} - 4288x^{9} + 118782x^{8} - 94248x^{7} - 59487x^{6} + 92762x^{5} - 14625x^{4} - 18074x^{3} + 7390x^{2} - 725x + 21$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(913))$	67
Old forms	16
Cusp forms $S_{2}(\Gamma_0(913))$	83
Eisenstein series $E_{2}(\Gamma_0(913))$	3
Modular forms $M_{2}(\Gamma_0(913))$	86

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(913)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(83))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.268 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, 11, 83, 913]: [0, 2, 8, 86] (certified at ell = 2147478731)
T = [(2, 1), (3, 1)]; h of degree 67 = 1 x 67 (phi(ord eps) x dim S_k^new); 7 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(913, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 913:2 1000

← 912.2.a · 914.2.a →