Atlas › Modular forms › Weight 2 › Level 380 › 380.2.a

Newspace 380.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(380))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
380.2.a.a	1	$\mathbb{Q}$	$q - q^{5} - 2q^{7} + O(q^{8})$	+ −	380a1 rank 1
380.2.a.b	1	$\mathbb{Q}$	$q + 2q^{3} - q^{5} + 2q^{7} + O(q^{8})$	+ +	380b1 rank 0
380.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{3} + 2q^{5} - 4q^{7} + O(q^{8})$	− +
380.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} + 2q^{5} + 4q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
380.2.a.a	$x$
380.2.a.b	$x - 68$
380.2.a.c	$x^{2} + 100x + 2492$
380.2.a.d	$x^{2} - 3468$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(380))$	6
Old forms	49
Cusp forms $S_{2}(\Gamma_0(380))$	55
Eisenstein series $E_{2}(\Gamma_0(380))$	11
Modular forms $M_{2}(\Gamma_0(380))$	66

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(380)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(20))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(76))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(95))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(190))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.258 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, 5, 10, 19, 20, 38, 76, 95, 190, 380]: [0, 1, 2, 1, 3, 2, 6, 7, 13, 12, 34, 66] (certified at ell = 2147483629)
T = [(3, 1), (7, 3), (11, 5), (13, 7), (17, 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(380, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 380:2 1000

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