Atlas › Modular forms › Weight 2 › Level 376 › 376.2.a

Newspace 376.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(376))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
376.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{3} - 4q^{5} + q^{7} + O(q^{8})$	−
376.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{3} - 2q^{5} - 3q^{7} + O(q^{8})$	+
376.2.a.c	4	degree 4	$4q - q^{3} + 4q^{5} - 3q^{7} + O(q^{8})$	+
376.2.a.d	4	degree 4	$4q + 3q^{3} + 5q^{7} + O(q^{8})$	−

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 $T$
376.2.a.a	$x^{2} + 13x + 41$
376.2.a.b	$x^{2} + 7x - 19$
376.2.a.c	$x^{4} - 11x^{3} - 69x^{2} + 554x + 1516$
376.2.a.d	$x^{4} - 3x^{3} - 161x^{2} + 742x - 554$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(376))$	12
Old forms	33
Cusp forms $S_{2}(\Gamma_0(376))$	45
Eisenstein series $E_{2}(\Gamma_0(376))$	7
Modular forms $M_{2}(\Gamma_0(376))$	52

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(376)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(94))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(188))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.293 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, 8, 47, 94, 188, 376]: [0, 1, 2, 3, 5, 14, 27, 52] (certified at ell = 2147483137)
T = [(3, 1), (5, 3)]; h of degree 12 = 1 x 12 (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(376, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 376:2 1000

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