Atlas › Modular forms › Weight 6 › Level 37 › 37.6.a

Newspace 37.6.a: $S_{6}^{\mathrm{new}}(\Gamma_0(37))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
37.6.a.a	7	degree 7	$7q - 8q^{2} - 47q^{3} + 106q^{4} - 64q^{5} - 141q^{6} - 293q^{7} + O(q^{8})$	+	$-1$
37.6.a.b	8	degree 8	$8q + 12q^{2} + 25q^{3} + 154q^{4} + 136q^{5} + 75q^{6} + 99q^{7} + O(q^{8})$	−	$+1$

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 $T$
37.6.a.a	$x^{7} + 55x^{6} + 492x^{5} - 13632x^{4} - 251662x^{3} - 1400444x^{2} - 2475561x + 351451$
37.6.a.b	$x^{8} - 37x^{7} - 916x^{6} + 42708x^{5} + 66376x^{4} - 13653342x^{3} + 74415411x^{2} + 1078175133x - 8412796410$

Dimensions

Newforms $S_{6}^{\mathrm{new}}(\Gamma_0(37))$	15
Old forms	0
Cusp forms $S_{6}(\Gamma_0(37))$	15
Eisenstein series $E_{6}(\Gamma_0(37))$	2
Modular forms $M_{6}(\Gamma_0(37))$	17

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.22 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, 17] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 15 = 1 x 15 (phi(ord eps) x dim S_k^new); 3 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, 6)

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

cargo run --release -p sagebrush-web --example atlas -- 37:6 1000

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