Atlas › Modular forms › Weight 4 › Level 157 › 157.4.a

Newspace 157.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(157))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
157.4.a.a	18	degree 18	$18q - 15q^{2} - 23q^{3} + 59q^{4} - 25q^{5} - 27q^{6} - 55q^{7} + O(q^{8})$	−	$-1$
157.4.a.b	21	degree 21	$21q + 11q^{2} + 25q^{3} + 95q^{4} + 15q^{5} + 9q^{6} + 29q^{7} + O(q^{8})$	+	$+1$

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 $T$
157.4.a.a	$x^{18} + 38x^{17} + 351x^{16} - 3064x^{15} - 60247x^{14} - 38558x^{13} + 3197723x^{12} + 9207351x^{11} - 70946691x^{10} - 262573383x^{9} + 576754029x^{8} + 2306434808x^{7} + 5702758x^{6} - 5299288274x^{5} - 4693966195x^{4} + 258235703x^{3} + 1912138744x^{2} + 815853509x + 106339027$
157.4.a.b	$x^{21} - 36x^{20} + 103x^{19} + 9878x^{18} - 92125x^{17} - 999898x^{16} + 14458373x^{15} + 40602847x^{14} - 1074332817x^{13} + 63480815x^{12} + 43959513911x^{11} - 61419049406x^{10} - 1019488111238x^{9} + 2093104121666x^{8} + 12933713966875x^{7} - 30437731639289x^{6} - 79578057589126x^{5} + 204565834404879x^{4} + 149500996778515x^{3} - 534638519740816x^{2} + 219035053957980x + 62554906820512$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(157))$	39
Old forms	0
Cusp forms $S_{4}(\Gamma_0(157))$	39
Eisenstein series $E_{4}(\Gamma_0(157))$	2
Modular forms $M_{4}(\Gamma_0(157))$	41

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.12 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, 157]: [1, 41] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 6 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(157, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 157:4 1000

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