Atlas › Modular forms › Weight 2 › Level 351 › 351.2.a

Newspace 351.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(351))$

Newforms

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

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
351.2.a.a	$x^{2} + 10x + 5$
351.2.a.b	$x^{2} + 16x + 51$
351.2.a.c	$x^{2} - 10x + 5$
351.2.a.d	$x^{2} - 16x + 51$
351.2.a.e	$x^{4} - 145x^{2} + 363$
351.2.a.f	$x^{4} - 195x^{2} + 475$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(351))$	16
Old forms	21
Cusp forms $S_{2}(\Gamma_0(351))$	37
Eisenstein series $E_{2}(\Gamma_0(351))$	11
Modular forms $M_{2}(\Gamma_0(351))$	48

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(351)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(39))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(117))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.203 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, 3, 9, 13, 27, 39, 117, 351]: [0, 1, 2, 1, 4, 6, 16, 44] (certified at ell = 2147483629)
T = [(2, 1), (5, 3)]; h of degree 16 = 1 x 16 (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(351, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 351:2 1000

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