Atlas › Modular forms › Weight 2 › Level 965 › 965.2.a

Newspace 965.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(965))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
965.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - q^{5} + 4q^{7} + O(q^{8})$	+ −	$+1$	965a1 rank 0
965.2.a.b	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{2} - q^{3} + 3q^{4} + 2q^{5} - 7q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
965.2.a.c	8	degree 8	$8q - 2q^{2} - 6q^{3} - 2q^{4} + 8q^{5} + 4q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
965.2.a.d	12	degree 12	$12q - q^{2} - 7q^{3} + 3q^{4} - 12q^{5} + q^{6} - 6q^{7} + O(q^{8})$	+ +	$-1$
965.2.a.e	20	degree 20	$20q + 3q^{2} + 11q^{3} + 31q^{4} - 20q^{5} - q^{6} + 10q^{7} + O(q^{8})$	+ −	$+1$
965.2.a.f	22	degree 22	$22q + q^{2} + 7q^{3} + 33q^{4} + 22q^{5} + 7q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$

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 $T$
965.2.a.a	$x - 1$
965.2.a.b	$x^{2} + 2x - 12$
965.2.a.c	$x^{8} + 20x^{7} + 82x^{6} - 533x^{5} - 3432x^{4} + 2807x^{3} + 26848x^{2} - 1943x - 32519$
965.2.a.d	$x^{12} + 22x^{11} + 23x^{10} - 2289x^{9} - 10188x^{8} + 79804x^{7} + 466594x^{6} - 1049250x^{5} - 7389521x^{4} + 3399795x^{3} + 35168341x^{2} + 11303278x - 9529324$
965.2.a.e	$x^{20} - 36x^{19} + 188x^{18} + 7331x^{17} - 88198x^{16} - 453055x^{15} + 10399619x^{14} + 662405x^{13} - 614747707x^{12} + 1157853741x^{11} + 21427118989x^{10} - 59977175395x^{9} - 469376631360x^{8} + 1478261624832x^{7} + 6585238456393x^{6} - 20002732812391x^{5} - 57987369580272x^{4} + 142682387131879x^{3} + 294705589616521x^{2} - 419252764273466x - 664160920628556$
965.2.a.f	$x^{22} - 22x^{21} - 245x^{20} + 8377x^{19} + 3776x^{18} - 1253036x^{17} + 3970069x^{16} + 95132492x^{15} - 490139586x^{14} - 3975292984x^{13} + 26791295104x^{12} + 90884949948x^{11} - 791828399229x^{10} - 965685744821x^{9} + 12940063235967x^{8} - 371389155035x^{7} - 107904077273163x^{6} + 100362307199160x^{5} + 338075843028277x^{4} - 627614497993563x^{3} + 236313456218277x^{2} + 91746193190482x - 51155939991844$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(965))$	65
Old forms	30
Cusp forms $S_{2}(\Gamma_0(965))$	95
Eisenstein series $E_{2}(\Gamma_0(965))$	3
Modular forms $M_{2}(\Gamma_0(965))$	98

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(965)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(193))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.302 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, 5, 193, 965]: [0, 1, 16, 98] (certified at ell = 2147483137)
T = [(2, 1), (3, 3)]; h of degree 65 = 1 x 65 (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(965, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 965:2 1000

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