Atlas › Modular forms › Weight 2 › Level 917 › 917.2.a

Newspace 917.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(917))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
917.2.a.a	14	degree 14	$14q - 5q^{2} - 6q^{3} + 9q^{4} + 3q^{5} - 4q^{6} - 14q^{7} + O(q^{8})$	+ +	$-1$
917.2.a.b	14	degree 14	$14q - 3q^{2} - 12q^{3} + 9q^{4} - 9q^{5} + 14q^{7} + O(q^{8})$	− −	$-1$
917.2.a.c	18	degree 18	$18q + 3q^{2} + 10q^{3} + 17q^{4} + 3q^{5} + 8q^{6} + 18q^{7} + O(q^{8})$	− +	$+1$
917.2.a.d	19	degree 19	$19q + 4q^{2} + 8q^{3} + 24q^{4} - 3q^{5} - 19q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
917.2.a.a	$x^{14} + 11x^{13} + 21x^{12} - 141x^{11} - 459x^{10} + 687x^{9} + 2910x^{8} - 2097x^{7} - 8732x^{6} + 5557x^{5} + 12811x^{4} - 10502x^{3} - 6373x^{2} + 8592x - 2279$
917.2.a.b	$x^{14} + 15x^{13} + 65x^{12} - 45x^{11} - 1003x^{10} - 1843x^{9} + 2506x^{8} + 9635x^{7} + 2096x^{6} - 14377x^{5} - 8229x^{4} + 7924x^{3} + 5581x^{2} - 1374x - 1051$
917.2.a.c	$x^{18} - 13x^{17} + 14x^{16} + 480x^{15} - 1955x^{14} - 3837x^{13} + 33844x^{12} - 24767x^{11} - 189091x^{10} + 367703x^{9} + 201463x^{8} - 1001582x^{7} + 532406x^{6} + 479977x^{5} - 497439x^{4} + 107538x^{3} + 3840x^{2} - 460x - 13$
917.2.a.d	$x^{19} - 12x^{18} + x^{17} + 446x^{16} - 919x^{15} - 6478x^{14} + 16889x^{13} + 50879x^{12} - 128088x^{11} - 253260x^{10} + 473350x^{9} + 802829x^{8} - 811034x^{7} - 1405689x^{6} + 529150x^{5} + 1178045x^{4} + 32772x^{3} - 346388x^{2} - 103127x - 8221$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(917))$	65
Old forms	22
Cusp forms $S_{2}(\Gamma_0(917))$	87
Eisenstein series $E_{2}(\Gamma_0(917))$	3
Modular forms $M_{2}(\Gamma_0(917))$	90

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(917)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(131))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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, 131, 917]: [0, 1, 12, 90] (certified at ell = 2147481571)
T = [(2, 1), (3, 1)]; h of degree 65 = 1 x 65 (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(917, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 917:2 1000

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