Atlas › Modular forms › Weight 2 › Level 313 › 313.2.a

Newspace 313.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(313))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
313.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} + 3q^{3} - q^{4} + 3q^{5} - q^{6} + 2q^{7} + O(q^{8})$	−	$+1$
313.2.a.b	11	degree 11	$11q - 8q^{2} - 8q^{3} + 10q^{4} - 6q^{5} + q^{6} - 14q^{7} + O(q^{8})$	+	$-1$
313.2.a.c	12	degree 12	$12q + 6q^{2} + q^{3} + 16q^{4} - q^{5} + 2q^{6} + 6q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
313.2.a.a	$x^{2} - 10x + 20$
313.2.a.b	$x^{11} + 32x^{10} + 322x^{9} + 391x^{8} - 11989x^{7} - 61376x^{6} - 6518x^{5} + 449731x^{4} + 325772x^{3} - 1097423x^{2} - 189982x + 264164$
313.2.a.c	$x^{12} - 9x^{11} - 197x^{10} + 1614x^{9} + 14584x^{8} - 102562x^{7} - 502992x^{6} + 2816990x^{5} + 7873681x^{4} - 30640686x^{3} - 41742920x^{2} + 68223533x - 19671289$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(313))$	25
Old forms	0
Cusp forms $S_{2}(\Gamma_0(313))$	25
Eisenstein series $E_{2}(\Gamma_0(313))$	1
Modular forms $M_{2}(\Gamma_0(313))$	26

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.175 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, 313]: [0, 26] (certified at ell = 2147482273)
T = [(2, 1), (3, 3)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 4 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(313, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 313:2 1000

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