Atlas › Modular forms › Weight 2 › Level 331 › 331.2.a

Newspace 331.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(331))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
331.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} - q^{4} + q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	+	$-1$	331a1 rank 1
331.2.a.b	3	degree 3	$3q - 2q^{2} - q^{3} + 6q^{4} - 6q^{5} - 10q^{6} + 3q^{7} + O(q^{8})$	+	$-1$
331.2.a.c	7	degree 7	$7q - 2q^{2} + 2q^{4} - 13q^{5} - q^{6} - 8q^{7} + O(q^{8})$	+	$-1$
331.2.a.d	16	degree 16	$16q + 3q^{2} - q^{3} + 15q^{4} + 20q^{5} + 3q^{6} + q^{7} + O(q^{8})$	−	$+1$

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 $T$
331.2.a.a	$x + 7$
331.2.a.b	$x^{3} + 5x^{2} - 13x - 1$
331.2.a.c	$x^{7} + 2x^{6} - 93x^{5} - 50x^{4} + 596x^{3} + 351x^{2} - 869x - 467$
331.2.a.d	$x^{16} - 334x^{14} + 234x^{13} + 43606x^{12} - 51063x^{11} - 2858635x^{10} + 4022824x^{9} + 101091696x^{8} - 146654194x^{7} - 1909455369x^{6} + 2546319514x^{5} + 17642915519x^{4} - 18803238363x^{3} - 66184242440x^{2} + 40528913396x + 82514262440$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(331))$	27
Old forms	0
Cusp forms $S_{2}(\Gamma_0(331))$	27
Eisenstein series $E_{2}(\Gamma_0(331))$	1
Modular forms $M_{2}(\Gamma_0(331))$	28

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.179 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, 331]: [0, 28] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 27 = 1 x 27 (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(331, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 331:2 1000

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