Atlas › Modular forms › Weight 2 › Level 363 › 363.2.a

Newspace 363.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(363))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
363.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + q^{7} + O(q^{8})$	+	363c1 rank 0
363.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} - 4q^{7} + O(q^{8})$	+	363a1 rank 0
363.2.a.c	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 2q^{4} + 4q^{5} - 2q^{6} - q^{7} + O(q^{8})$	+	363b1 rank 0
363.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} - 2q^{3} + 3q^{4} + q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	+
363.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 2q^{3} - q^{4} - 3q^{5} - q^{6} - 6q^{7} + O(q^{8})$	−
363.2.a.f	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{4} - 6q^{5} + O(q^{8})$	+
363.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} + 6q^{4} + 4q^{5} + O(q^{8})$	−
363.2.a.h	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} + 2q^{3} - q^{4} - 3q^{5} + q^{6} + 6q^{7} + O(q^{8})$	−
363.2.a.i	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} - 2q^{3} + 3q^{4} + q^{5} - 3q^{6} + 2q^{7} + O(q^{8})$	+
363.2.a.j	4	degree 4	$4q + 4q^{3} + 6q^{4} + 2q^{5} + O(q^{8})$	−

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 $T$
363.2.a.a	$x - 10$
363.2.a.b	$x + 7$
363.2.a.c	$x - 14$
363.2.a.d	$x^{2} - 20$
363.2.a.e	$x^{2} + 10x + 20$
363.2.a.f	$x^{2} + 18x + 78$
363.2.a.g	$x^{2} - 12x + 31$
363.2.a.h	$x^{2} + 8x - 4$
363.2.a.i	$x^{2} - 6x + 4$
363.2.a.j	$x^{4} - 6x^{3} - 142x^{2} + 552x + 4504$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(363))$	19
Old forms	14
Cusp forms $S_{2}(\Gamma_0(363))$	33
Eisenstein series $E_{2}(\Gamma_0(363))$	23
Modular forms $M_{2}(\Gamma_0(363))$	56

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(363)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(33))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(121))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.318 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, 11, 33, 121, 363]: [0, 1, 2, 6, 12, 46] (certified at ell = 2147482921)
T = [(2, 1), (5, 3)]; h of degree 19 = 1 x 19 (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(363, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 363:2 1000

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