Atlas › Modular forms › Weight 2 › Level 362 › 362.2.a

Newspace 362.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(362))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
362.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$	362a1 rank 1
362.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$	362b1 rank 1
362.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
362.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + O(q^{8})$	− +	$+1$
362.2.a.e	5	degree 5	$5q - 5q^{2} + 4q^{3} + 5q^{4} - 4q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$
362.2.a.f	5	degree 5	$5q + 5q^{2} + 5q^{4} - q^{5} + 6q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
362.2.a.a	$x - 5$
362.2.a.b	$x + 7$
362.2.a.c	$x^{2} + 5x + 5$
362.2.a.d	$x^{2} - 14x + 41$
362.2.a.e	$x^{5} - 4x^{4} - 188x^{3} + 803x^{2} + 4325x - 7949$
362.2.a.f	$x^{5} + 3x^{4} - 166x^{3} - 192x^{2} + 3590x + 7415$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(362))$	16
Old forms	28
Cusp forms $S_{2}(\Gamma_0(362))$	44
Eisenstein series $E_{2}(\Gamma_0(362))$	3
Modular forms $M_{2}(\Gamma_0(362))$	47

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(362)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(181))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.195 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, 181, 362]: [0, 1, 15, 47] (certified at ell = 2147482801)
T = [(3, 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(362, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 362:2 1000

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