Atlas › Modular forms › Weight 4 › Level 37 › 37.4.a

Newspace 37.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(37))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
37.4.a.a	4	degree 4	$4q - 6q^{2} - 11q^{3} + 6q^{4} - 29q^{5} - 27q^{6} - 32q^{7} + O(q^{8})$	−	$-1$
37.4.a.b	5	degree 5	$5q + 4q^{2} + 13q^{3} + 18q^{4} + 11q^{5} + 9q^{6} + 24q^{7} + O(q^{8})$	+	$+1$

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 $T$
37.4.a.a	$x^{4} + 17x^{3} + 98x^{2} + 231x + 187$
37.4.a.b	$x^{5} - 17x^{4} + 16x^{3} + 955x^{2} - 4905x + 6008$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(37))$	9
Old forms	0
Cusp forms $S_{4}(\Gamma_0(37))$	9
Eisenstein series $E_{4}(\Gamma_0(37))$	2
Modular forms $M_{4}(\Gamma_0(37))$	11

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.092 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, 37]: [1, 11] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 2 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(37, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 37:4 1000

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