Atlas › Modular forms › Weight 2 › Level 397 › 397.2.a

Newspace 397.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(397))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
397.2.a.a	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} - 4q^{5} + 6q^{7} + O(q^{8})$	+	$-1$
397.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 4q^{3} + 2q^{4} - 2q^{7} + O(q^{8})$	−	$+1$
397.2.a.c	5	degree 5	$5q + 5q^{3} + 2q^{4} - 2q^{5} + 12q^{6} + q^{7} + O(q^{8})$	−	$+1$
397.2.a.d	10	degree 10	$10q + 7q^{2} + 13q^{4} + 7q^{5} - 11q^{6} + 10q^{7} + O(q^{8})$	−	$+1$
397.2.a.e	13	degree 13	$13q - 7q^{2} - 11q^{3} + 13q^{4} - 5q^{5} - q^{6} - 17q^{7} + O(q^{8})$	+	$-1$

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 $T$
397.2.a.a	$x^{2} + 2x - 1$
397.2.a.b	$x^{2} - 14x + 41$
397.2.a.c	$x^{5} - 15x^{4} - 6x^{3} + 658x^{2} - 779x - 3475$
397.2.a.d	$x^{10} - 7x^{9} - 130x^{8} + 877x^{7} + 5880x^{6} - 38417x^{5} - 110142x^{4} + 680164x^{3} + 810830x^{2} - 3935080x - 2418033$
397.2.a.e	$x^{13} + 40x^{12} + 518x^{11} + 843x^{10} - 33730x^{9} - 221300x^{8} + 310432x^{7} + 6226961x^{6} + 11212927x^{5} - 36930628x^{4} - 158098473x^{3} - 210784573x^{2} - 115379355x - 21696887$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(397))$	32
Old forms	0
Cusp forms $S_{2}(\Gamma_0(397))$	32
Eisenstein series $E_{2}(\Gamma_0(397))$	1
Modular forms $M_{2}(\Gamma_0(397))$	33

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.186 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, 397]: [0, 33] (certified at ell = 2147483053)
T = [(2, 1), (3, 3)]; h of degree 32 = 1 x 32 (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(397, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 397:2 1000

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