Atlas › Modular forms › Weight 2 › Level 361 › 361.2.a

Newspace 361.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(361))$

Newforms

Label	Dim	Coefficient field	Trace form	CM	Elliptic curve
361.2.a.a	1	$\mathbb{Q}$	$q - 2q^{4} - q^{5} + 3q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-19})$	361a1 rank 1
361.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} - 2q^{4} + 3q^{5} - q^{7} + O(q^{8})$		361b1 rank 0
361.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - 3q^{3} - q^{4} + 2q^{5} - q^{6} + 6q^{7} + O(q^{8})$
361.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - 4q^{3} + 6q^{4} + q^{5} - 2q^{7} + O(q^{8})$
361.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q + 4q^{3} + 6q^{4} + q^{5} - 2q^{7} + O(q^{8})$
361.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} + 3q^{3} - q^{4} + 2q^{5} - q^{6} + 6q^{7} + O(q^{8})$
361.2.a.g	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} - 3q^{5} + 6q^{6} + O(q^{8})$
361.2.a.h	3	degree 3	$3q + 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} + 6q^{6} + O(q^{8})$
361.2.a.i	4	degree 4	$4q + 2q^{4} - 4q^{5} - 10q^{6} - 8q^{7} + O(q^{8})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
361.2.a.a	$x$
361.2.a.b	$x - 6$
361.2.a.c	$x^{2} + 10x + 20$
361.2.a.d	$x^{2} + 12x + 31$
361.2.a.e	$x^{2} - 12x + 31$
361.2.a.f	$x^{2} - 10x + 20$
361.2.a.g	$x^{3} + 12x^{2} + 9x - 111$
361.2.a.h	$x^{3} - 12x^{2} + 9x + 111$
361.2.a.i	$x^{4} - 20x^{2} + 80$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(361))$	20
Old forms	2
Cusp forms $S_{2}(\Gamma_0(361))$	22
Eisenstein series $E_{2}(\Gamma_0(361))$	19
Modular forms $M_{2}(\Gamma_0(361))$	41

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(361)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.179 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, 19, 361]: [0, 2, 32] (certified at ell = 2147483323)
T = [(2, 1), (3, 3)]; h of degree 20 = 1 x 20 (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(361, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 361:2 1000

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