Atlas › Modular forms › Weight 2 › Level 356 › 356.2.a

Newspace 356.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(356))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
356.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - q^{5} + O(q^{8})$	−			356a1 rank 1
356.2.a.b	7	degree 7	$7q + q^{3} + 3q^{5} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
356.2.a.a	$x + 41$
356.2.a.b	$x^{7} - 183x^{6} + 2162x^{5} + 1028640x^{4} - 46258905x^{3} + 146798311x^{2} + 9322528422x - 56115796062$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(356))$	8
Old forms	35
Cusp forms $S_{2}(\Gamma_0(356))$	43
Eisenstein series $E_{2}(\Gamma_0(356))$	5
Modular forms $M_{2}(\Gamma_0(356))$	48

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(356)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(89))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(178))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.212 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, 2, 4, 89, 178, 356]: [0, 1, 2, 8, 24, 48] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 8 = 1 x 8 (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(356, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 356:2 1000

← 355.2.a · 357.2.a →