Atlas › Modular forms › Weight 2 › Level 371 › 371.2.a

Newspace 371.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(371))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
371.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{6} - q^{7} + O(q^{8})$	+ +	$-1$	371a1 rank 1
371.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + 3q^{5} + q^{7} + O(q^{8})$	− +	$+1$	371b1 rank 0
371.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - q^{3} - q^{4} - 3q^{5} + 3q^{6} + 2q^{7} + O(q^{8})$	− −	$-1$
371.2.a.d	3	degree 3	$3q + 2q^{4} - 5q^{5} - 8q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
371.2.a.e	9	degree 9	$9q + 3q^{3} + 12q^{4} + 9q^{5} + q^{6} - 9q^{7} + O(q^{8})$	+ −	$+1$
371.2.a.f	11	degree 11	$11q - q^{2} - q^{3} + 19q^{4} + 2q^{5} + q^{6} + 11q^{7} + O(q^{8})$	− +	$+1$

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 $T$
371.2.a.a	$x + 2$
371.2.a.b	$x - 2$
371.2.a.c	$x^{2} + 4x - 16$
371.2.a.d	$x^{3} - 16x + 8$
371.2.a.e	$x^{9} - 9x^{8} - 153x^{7} + 1285x^{6} + 7772x^{5} - 54264x^{4} - 169764x^{3} + 564444x^{2} + 2133496x + 1640048$
371.2.a.f	$x^{11} + 4x^{10} - 254x^{9} - 616x^{8} + 23141x^{7} + 29744x^{6} - 964668x^{5} - 306948x^{4} + 18681776x^{3} - 10675088x^{2} - 136958848x + 189864960$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(371))$	27
Old forms	8
Cusp forms $S_{2}(\Gamma_0(371))$	35
Eisenstein series $E_{2}(\Gamma_0(371))$	3
Modular forms $M_{2}(\Gamma_0(371))$	38

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(371)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(53))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.192 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, 7, 53, 371]: [0, 1, 5, 38] (certified at ell = 2147483053)
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(371, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 371:2 1000

← 370.2.a · 372.2.a →