Atlas › Modular forms › Weight 2 › Level 364 › 364.2.a

Newspace 364.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(364))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
364.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + q^{5} - q^{7} + O(q^{8})$	+ −	364b1 rank 1
364.2.a.b	1	$\mathbb{Q}$	$q - 3q^{5} + q^{7} + O(q^{8})$	− +	364a1 rank 1
364.2.a.c	2	$\mathbb{Q}(\sqrt{6})$	$2q - 2q^{5} - 2q^{7} + O(q^{8})$	+ +
364.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{17} + 7T_{19} + 9T_{23}$ 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 $T$
364.2.a.a	$x + 79$
364.2.a.b	$x + 99$
364.2.a.c	$x^{2} - 18x - 783$
364.2.a.d	$x^{2} - 114x + 3057$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(364))$	6
Old forms	45
Cusp forms $S_{2}(\Gamma_0(364))$	51
Eisenstein series $E_{2}(\Gamma_0(364))$	11
Modular forms $M_{2}(\Gamma_0(364))$	62

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(364)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(91))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(182))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.38 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, 7, 13, 14, 26, 28, 52, 91, 182, 364]: [0, 1, 2, 1, 1, 4, 5, 7, 10, 10, 32, 62] (certified at ell = 2147483629)
T = [(3, 1), (5, 3), (17, 5), (19, 7), (23, 9)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 2 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(364, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 364:2 1000

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