Atlas › Modular forms › Weight 2 › Level 961 › 961.2.a

Newspace 961.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(961))$

Newforms

Label	Dim	Coefficient field	Trace form	CM
961.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 6q^{6} + 2q^{7} + O(q^{8})$
961.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{4} + 8q^{7} + O(q^{8})$
961.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 6q^{6} + 2q^{7} + O(q^{8})$
961.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - 2q^{3} - q^{4} - 3q^{5} - q^{6} + 6q^{7} + O(q^{8})$
961.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} + 2q^{3} - q^{4} - 3q^{5} + q^{6} + 6q^{7} + O(q^{8})$
961.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} + 2q^{3} - q^{4} + 2q^{5} + 6q^{6} - 4q^{7} + O(q^{8})$
961.2.a.g	3	degree 3	$3q + 6q^{4} + O(q^{8})$	$\mathbb{Q}(\sqrt{-31})$
961.2.a.h	4	degree 4	$4q + 6q^{2} + 6q^{4} - 4q^{7} + O(q^{8})$
961.2.a.i	8	degree 8	$8q + 2q^{2} - 3q^{3} + 8q^{4} + 3q^{5} - 11q^{6} - 2q^{7} + O(q^{8})$
961.2.a.j	8	degree 8	$8q + 2q^{2} + 3q^{3} + 8q^{4} + 3q^{5} + 11q^{6} - 2q^{7} + O(q^{8})$
961.2.a.k	12	degree 12	$12q + 16q^{4} + 8q^{5} + 8q^{7} + O(q^{8})$
961.2.a.l	16	degree 16	$16q - 8q^{2} + 8q^{4} - 16q^{5} - 16q^{7} + O(q^{8})$

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 $T$
961.2.a.a	$x^{2} + 8x - 16$
961.2.a.b	$x^{2} + 2x - 71$
961.2.a.c	$x^{2} - 4x - 4$
961.2.a.d	$x^{2} + 5x + 5$
961.2.a.e	$x^{2} - 7x + 11$
961.2.a.f	$x^{2} - 7x - 49$
961.2.a.g	$x^{3} - 6x - 1$
961.2.a.h	$x^{4} - 6x^{3} - 43x^{2} + 246x + 1$
961.2.a.i	$x^{8} + 7x^{7} - 85x^{6} - 628x^{5} + 1414x^{4} + 14388x^{3} + 9870x^{2} - 41967x - 15399$
961.2.a.j	$x^{8} - 11x^{7} - 115x^{6} + 1106x^{5} + 5824x^{4} - 30534x^{3} - 145920x^{2} + 75069x + 568881$
961.2.a.k	$x^{12} - 272x^{10} + 68x^{9} + 26958x^{8} - 3040x^{7} - 1223078x^{6} - 649712x^{5} + 26378238x^{4} + 39913140x^{3} - 209414112x^{2} - 600150944x - 405188639$
961.2.a.l	$x^{16} + 8x^{15} - 204x^{14} - 1360x^{13} + 17378x^{12} + 80344x^{11} - 822272x^{10} - 1839688x^{9} + 22341241x^{8} + 1996200x^{7} - 308643296x^{6} + 503560408x^{5} + 1353160274x^{4} - 5085997584x^{3} + 5291101324x^{2} - 1021541192x - 859286303$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(961))$	63
Old forms	4
Cusp forms $S_{2}(\Gamma_0(961))$	67
Eisenstein series $E_{2}(\Gamma_0(961))$	31
Modular forms $M_{2}(\Gamma_0(961))$	98

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(961)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 31, 961]: [0, 3, 83] (certified at ell = 2147477881)
T = [(2, 1), (3, 3)]; h of degree 63 = 1 x 63 (phi(ord eps) x dim S_k^new); 9 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(961, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 961:2 1000

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