Atlas › Modular forms › Weight 4 › Level 131 › 131.4.a

Newspace 131.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(131))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
131.4.a.a	11	degree 11	$11q - 3q^{2} - 11q^{3} + 11q^{4} - 27q^{5} - 22q^{6} - 32q^{7} + O(q^{8})$	−	$-1$
131.4.a.b	21	degree 21	$21q + 3q^{2} + 7q^{3} + 111q^{4} + 33q^{5} + 26q^{6} + 38q^{7} + O(q^{8})$	+	$+1$

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 $T$
131.4.a.a	$x^{11} + 14x^{10} - 85x^{9} - 1361x^{8} + 2076x^{7} + 34999x^{6} - 46085x^{5} - 269590x^{4} + 485437x^{3} + 176877x^{2} - 452668x - 47687$
131.4.a.b	$x^{21} - 10x^{20} - 541x^{19} + 5099x^{18} + 123175x^{17} - 1057095x^{16} - 15733206x^{15} + 117705933x^{14} + 1254427992x^{13} - 7773513722x^{12} - 65153242023x^{11} + 315470225012x^{10} + 2225169017642x^{9} - 7820379611531x^{8} - 49330633034639x^{7} + 111365682870920x^{6} + 682586911022760x^{5} - 738157256977079x^{4} - 5376488235466859x^{3} + 27196966688368x^{2} + 18564728852327322x + 16972856964400193$

Orbit

Characteristic polynomial of

T

131.4.a.a

x^{11} + 14x^{10} - 85x^{9} - 1361x^{8} + 2076x^{7} + 34999x^{6} - 46085x^{5} - 269590x^{4} + 485437x^{3} + 176877x^{2} - 452668x - 47687

131.4.a.b

x^{21} - 10x^{20} - 541x^{19} + 5099x^{18} + 123175x^{17} - 1057095x^{16} - 15733206x^{15} + 117705933x^{14} + 1254427992x^{13} - 7773513722x^{12} - 65153242023x^{11} + 315470225012x^{10} + 2225169017642x^{9} - 7820379611531x^{8} - 49330633034639x^{7} + 111365682870920x^{6} + 682586911022760x^{5} - 738157256977079x^{4} - 5376488235466859x^{3} + 27196966688368x^{2} + 18564728852327322x + 16972856964400193

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(131))$	32
Old forms	0
Cusp forms $S_{4}(\Gamma_0(131))$	32
Eisenstein series $E_{4}(\Gamma_0(131))$	2
Modular forms $M_{4}(\Gamma_0(131))$	34

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.11 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, 131]: [1, 34] (certified at ell = 2147482481)
T = [(2, 1), (3, 1)]; h of degree 32 = 1 x 32 (phi(ord eps) x dim S_k^new); 5 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(131, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 131:4 1000

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