Atlas › Modular forms › Weight 8 › Level 37 › 37.8.a

Newspace 37.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(37))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
37.8.a.a	10	degree 10	$10q - 24q^{2} - 95q^{3} + 602q^{4} - 624q^{5} - 777q^{6} - 501q^{7} + O(q^{8})$	−	$-1$
37.8.a.b	11	degree 11	$11q + 16q^{2} + 121q^{3} + 794q^{4} + 376q^{5} + 519q^{6} + 2243q^{7} + O(q^{8})$	+	$+1$

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 $T$
37.8.a.a	$x^{10} + 119x^{9} - 7109x^{8} - 996295x^{7} + 17503960x^{6} + 2689381514x^{5} - 15779310925x^{4} - 2532238654253x^{3} + 610609057119x^{2} + 620523208892907x + 2449613912536098$
37.8.a.b	$x^{11} - 137x^{10} - 9177x^{9} + 1544077x^{8} + 28512590x^{7} - 5745282384x^{6} - 49814019603x^{5} + 7581539205123x^{4} + 76274251969145x^{3} - 2199792138785719x^{2} - 10850964339267476x + 169022040552063416$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(37))$	21
Old forms	0
Cusp forms $S_{8}(\Gamma_0(37))$	21
Eisenstein series $E_{8}(\Gamma_0(37))$	2
Modular forms $M_{8}(\Gamma_0(37))$	23

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 23] (certified at ell = 2147483629)
T = [(2, 1), (3, 1)]; h of degree 21 = 1 x 21 (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(37, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 37:8 1000

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