Atlas › Modular forms › Weight 4 › Level 137 › 137.4.a

Newspace 137.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(137))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
137.4.a.a	15	degree 15	$15q - 6q^{2} - 18q^{3} + 44q^{4} - 20q^{5} - 55q^{6} - 122q^{7} + O(q^{8})$	−	$-1$
137.4.a.b	19	degree 19	$19q + 6q^{2} + 18q^{3} + 92q^{4} + 10q^{5} + 5q^{6} + 130q^{7} + O(q^{8})$	+	$+1$

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 $T$
137.4.a.a	$x^{15} + 24x^{14} + 10x^{13} - 2945x^{12} - 10142x^{11} + 116511x^{10} + 383221x^{9} - 2352344x^{8} - 4675588x^{7} + 25331967x^{6} + 11100081x^{5} - 114567032x^{4} + 85099013x^{3} + 29352834x^{2} - 10605996x - 2664063$
137.4.a.b	$x^{19} - 24x^{18} - 198x^{17} + 8079x^{16} + 256x^{15} - 1107317x^{14} + 2827761x^{13} + 79643772x^{12} - 294683886x^{11} - 3224481899x^{10} + 13697286267x^{9} + 73042813204x^{8} - 328674738677x^{7} - 858677051430x^{6} + 4018694634960x^{5} + 4227866187587x^{4} - 22551343878518x^{3} - 3528554239274x^{2} + 42735267406066x - 10511080394762$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(137))$	34
Old forms	0
Cusp forms $S_{4}(\Gamma_0(137))$	34
Eisenstein series $E_{4}(\Gamma_0(137))$	2
Modular forms $M_{4}(\Gamma_0(137))$	36

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.111 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, 137]: [1, 36] (certified at ell = 2147483249)
T = [(2, 1), (3, 1)]; h of degree 34 = 1 x 34 (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(137, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 137:4 1000

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