Atlas › Modular forms › Weight 2 › Level 311 › 311.2.a

Newspace 311.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(311))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
311.2.a.a	4	degree 4	$4q - q^{2} - 2q^{3} - q^{4} - q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	+	$-1$
311.2.a.b	22	degree 22	$22q + 2q^{2} + 2q^{3} + 30q^{4} + q^{5} + 2q^{6} + 8q^{7} + O(q^{8})$	−	$+1$

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 $T$
311.2.a.a	$x^{4} + 3x^{3} - 4x - 1$
311.2.a.b	$x^{22} - 4x^{21} - 83x^{20} + 344x^{19} + 2721x^{18} - 11867x^{17} - 45429x^{16} + 214882x^{15} + 412401x^{14} - 2251279x^{13} - 1928513x^{12} + 14125266x^{11} + 2893612x^{10} - 52649821x^{9} + 11137652x^{8} + 110674157x^{7} - 51561324x^{6} - 117814489x^{5} + 68112425x^{4} + 54704945x^{3} - 31139582x^{2} - 7575637x + 3738089$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(311))$	26
Old forms	0
Cusp forms $S_{2}(\Gamma_0(311))$	26
Eisenstein series $E_{2}(\Gamma_0(311))$	1
Modular forms $M_{2}(\Gamma_0(311))$	27

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.174 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, 311]: [0, 27] (certified at ell = 2147479121)
T = [(2, 1), (3, 1)]; h of degree 26 = 1 x 26 (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(311, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 311:2 1000

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