Atlas › Modular forms › Weight 4 › Level 149 › 149.4.a

Newspace 149.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(149))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
149.4.a.a	15	degree 15	$15q - 4q^{2} - 22q^{3} + 26q^{4} - 16q^{5} - 41q^{6} - 46q^{7} + O(q^{8})$	−	$-1$
149.4.a.b	22	degree 22	$22q + 2q^{2} + 20q^{3} + 110q^{4} + 24q^{5} + 43q^{6} + 66q^{7} + O(q^{8})$	+	$+1$

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 $T$
149.4.a.a	$x^{15} + 26x^{14} + 67x^{13} - 2924x^{12} - 17617x^{11} + 101980x^{10} + 789982x^{9} - 1161723x^{8} - 12275469x^{7} + 2561507x^{6} + 66252288x^{5} + 7965165x^{4} - 68729554x^{3} + 1448928x^{2} + 13990940x + 167351$
149.4.a.b	$x^{22} - 22x^{21} - 372x^{20} + 10506x^{19} + 43744x^{18} - 2029364x^{17} - 894806x^{16} + 214287269x^{15} - 235787718x^{14} - 13810039198x^{13} + 24479763342x^{12} + 566443787816x^{11} - 1134633521632x^{10} - 14814151094368x^{9} + 29228120004077x^{8} + 238405578027866x^{7} - 421507451813190x^{6} - 2151388065845448x^{5} + 3067239610268564x^{4} + 8536701721305100x^{3} - 8156429562611558x^{2} - 3148897443538255x + 363166899029362$

Orbit

Characteristic polynomial of

T

149.4.a.a

x^{15} + 26x^{14} + 67x^{13} - 2924x^{12} - 17617x^{11} + 101980x^{10} + 789982x^{9} - 1161723x^{8} - 12275469x^{7} + 2561507x^{6} + 66252288x^{5} + 7965165x^{4} - 68729554x^{3} + 1448928x^{2} + 13990940x + 167351

149.4.a.b

x^{22} - 22x^{21} - 372x^{20} + 10506x^{19} + 43744x^{18} - 2029364x^{17} - 894806x^{16} + 214287269x^{15} - 235787718x^{14} - 13810039198x^{13} + 24479763342x^{12} + 566443787816x^{11} - 1134633521632x^{10} - 14814151094368x^{9} + 29228120004077x^{8} + 238405578027866x^{7} - 421507451813190x^{6} - 2151388065845448x^{5} + 3067239610268564x^{4} + 8536701721305100x^{3} - 8156429562611558x^{2} - 3148897443538255x + 363166899029362

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(149))$	37
Old forms	0
Cusp forms $S_{4}(\Gamma_0(149))$	37
Eisenstein series $E_{4}(\Gamma_0(149))$	2
Modular forms $M_{4}(\Gamma_0(149))$	39

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.115 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, 149]: [1, 39] (certified at ell = 2147481629)
T = [(2, 1), (3, 1)]; h of degree 37 = 1 x 37 (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(149, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 149:4 1000

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