Atlas › Modular forms › Weight 2 › Level 949 › 949.2.a

Newspace 949.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(949))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
949.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 3q^{3} - q^{4} + 4q^{5} - 4q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$
949.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} + 3q^{3} + 3q^{4} + 2q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$
949.2.a.c	16	degree 16	$16q - 3q^{2} - 11q^{3} + 11q^{4} - 6q^{5} - 4q^{6} + q^{7} + O(q^{8})$	− −	$-1$
949.2.a.d	16	degree 16	$16q - q^{2} + 3q^{3} + 17q^{4} + 4q^{5} + 8q^{6} - 3q^{7} + O(q^{8})$	− +	$+1$
949.2.a.e	18	degree 18	$18q - 3q^{2} - 3q^{3} + 17q^{4} - 12q^{5} - 14q^{6} - 7q^{7} + O(q^{8})$	+ +	$-1$
949.2.a.f	19	degree 19	$19q + 4q^{2} + 5q^{3} + 20q^{4} + 8q^{5} + 8q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
949.2.a.a	$x^{2} - 8x + 11$
949.2.a.b	$x^{2} - 12x + 31$
949.2.a.c	$x^{16} + 36x^{15} + 382x^{14} - 767x^{13} - 38262x^{12} - 147275x^{11} + 1110613x^{10} + 7825688x^{9} - 9490000x^{8} - 150093005x^{7} - 49926604x^{6} + 1333714352x^{5} + 1048154170x^{4} - 5488731330x^{3} - 4473869485x^{2} + 7661253473x + 6993036273$
949.2.a.d	$x^{16} - 8x^{15} - 318x^{14} + 2197x^{13} + 40434x^{12} - 225803x^{11} - 2680125x^{10} + 10924384x^{9} + 100144198x^{8} - 252344787x^{7} - 2083327704x^{6} + 2322460722x^{5} + 21731067424x^{4} - 1640010450x^{3} - 87358716685x^{2} - 23209973263x + 102132288983$
949.2.a.e	$x^{18} + 12x^{17} - 241x^{16} - 2885x^{15} + 23440x^{14} + 265096x^{13} - 1280687x^{12} - 12261903x^{11} + 43260233x^{10} + 305704839x^{9} - 870775120x^{8} - 4044975911x^{7} + 9024986388x^{6} + 26953918266x^{5} - 36280350729x^{4} - 77362220937x^{3} + 6777053538x^{2} + 24969017119x - 5566743311$
949.2.a.f	$x^{19} - 19x^{18} - 237x^{17} + 6282x^{16} + 9903x^{15} - 774724x^{14} + 1382051x^{13} + 45499740x^{12} - 142285086x^{11} - 1411591966x^{10} + 5019648875x^{9} + 24120596351x^{8} - 80189773201x^{7} - 220014454622x^{6} + 566955717513x^{5} + 886550975000x^{4} - 1684702395733x^{3} - 1084862432689x^{2} + 1496400510370x + 151159350959$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(949))$	73
Old forms	10
Cusp forms $S_{2}(\Gamma_0(949))$	83
Eisenstein series $E_{2}(\Gamma_0(949))$	3
Modular forms $M_{2}(\Gamma_0(949))$	86

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(949)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(73))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.291 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, 13, 73, 949]: [0, 1, 6, 86] (certified at ell = 2147482873)
T = [(2, 1), (3, 3)]; h of degree 73 = 1 x 73 (phi(ord eps) x dim S_k^new); 10 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(949, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 949:2 1000

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