Atlas › Modular forms › Weight 2 › Level 353 › 353.2.a

Newspace 353.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(353))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
353.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} - q^{4} + 2q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	−	$+1$	353a1 rank 0
353.2.a.b	3	degree 3	$3q + q^{2} + 3q^{3} + 7q^{4} + 2q^{5} + 6q^{6} + 2q^{7} + O(q^{8})$	−	$+1$
353.2.a.c	11	degree 11	$11q - 5q^{2} - 5q^{3} + 5q^{4} - 4q^{5} - 2q^{6} - 25q^{7} + O(q^{8})$	+	$-1$
353.2.a.d	14	degree 14	$14q + 4q^{2} - 2q^{3} + 16q^{4} - 4q^{5} - 8q^{6} + 29q^{7} + O(q^{8})$	−	$+1$

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 $T$
353.2.a.a	$x - 1$
353.2.a.b	$x^{3} - 4x^{2} - 10x + 41$
353.2.a.c	$x^{11} + 10x^{10} + 19x^{9} - 69x^{8} - 200x^{7} + 99x^{6} + 555x^{5} + 156x^{4} - 474x^{3} - 302x^{2} + 39x + 45$
353.2.a.d	$x^{14} - 2x^{13} - 40x^{12} + 74x^{11} + 532x^{10} - 926x^{9} - 3025x^{8} + 4996x^{7} + 7374x^{6} - 11457x^{5} - 6816x^{4} + 9682x^{3} + 2239x^{2} - 2219x - 263$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(353))$	29
Old forms	0
Cusp forms $S_{2}(\Gamma_0(353))$	29
Eisenstein series $E_{2}(\Gamma_0(353))$	1
Modular forms $M_{2}(\Gamma_0(353))$	30

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.181 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, 353]: [0, 30] (certified at ell = 2147480897)
T = [(2, 1), (3, 1)]; h of degree 29 = 1 x 29 (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(353, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 353:2 1000

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