Atlas › Modular forms › Weight 2 › Level 375 › 375.2.a

Newspace 375.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(375))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
375.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} + 2q^{3} + 3q^{4} - 3q^{6} - 5q^{7} + O(q^{8})$	−
375.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - 2q^{3} - q^{4} + q^{6} - q^{7} + O(q^{8})$	+
375.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} + 2q^{3} - q^{4} + q^{6} + q^{7} + O(q^{8})$	−
375.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} - 2q^{3} + 3q^{4} - 3q^{6} + 5q^{7} + O(q^{8})$	+
375.2.a.e	4	degree 4	$4q - 3q^{2} - 4q^{3} + 7q^{4} + 3q^{6} + 4q^{7} + O(q^{8})$	+
375.2.a.f	4	degree 4	$4q + 3q^{2} + 4q^{3} + 7q^{4} + 3q^{6} - 4q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{7}$ 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 $T$
375.2.a.a	$x^{2} + 18x + 76$
375.2.a.b	$x^{2} + 4x - 16$
375.2.a.c	$x^{2} - 4x - 16$
375.2.a.d	$x^{2} - 18x + 76$
375.2.a.e	$x^{4} - 9x^{3} - 177x^{2} + 907x + 9479$
375.2.a.f	$x^{4} + 9x^{3} - 177x^{2} - 907x + 9479$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(375))$	16
Old forms	25
Cusp forms $S_{2}(\Gamma_0(375))$	41
Eisenstein series $E_{2}(\Gamma_0(375))$	19
Modular forms $M_{2}(\Gamma_0(375))$	60

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(375)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(75))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(125))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.322 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, 5, 15, 25, 75, 125, 375]: [0, 1, 1, 4, 3, 12, 13, 52] (certified at ell = 2147482801)
T = [(2, 1), (7, 3)]; h of degree 16 = 1 x 16 (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(375, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 375:2 1000

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