Atlas › Modular forms › Weight 4 › Level 75 › 75.4.a

Newspace 75.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(75))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
75.4.a.a	1	$\mathbb{Q}$	$q - 3q^{2} + 3q^{3} + q^{4} - 9q^{6} - 20q^{7} + O(q^{8})$	−
75.4.a.b	1	$\mathbb{Q}$	$q - q^{2} - 3q^{3} - 7q^{4} + 3q^{6} + 24q^{7} + O(q^{8})$	+
75.4.a.c	2	$\mathbb{Q}(\sqrt{41})$	$2q - 3q^{2} - 6q^{3} + 9q^{4} + 9q^{6} - 6q^{7} + O(q^{8})$	+
75.4.a.d	2	$\mathbb{Q}(\sqrt{19})$	$2q - 2q^{2} + 6q^{3} + 24q^{4} - 6q^{6} + 26q^{7} + O(q^{8})$	−
75.4.a.e	2	$\mathbb{Q}(\sqrt{19})$	$2q + 2q^{2} - 6q^{3} + 24q^{4} - 6q^{6} - 26q^{7} + O(q^{8})$	+
75.4.a.f	2	$\mathbb{Q}(\sqrt{41})$	$2q + 3q^{2} + 6q^{3} + 9q^{4} + 9q^{6} + 6q^{7} + O(q^{8})$	−

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 $T$
75.4.a.a	$x + 63$
75.4.a.b	$x - 71$
75.4.a.c	$x^{2} + 21x - 2852$
75.4.a.d	$x^{2} - 76x - 1767$
75.4.a.e	$x^{2} + 76x - 1767$
75.4.a.f	$x^{2} - 21x - 2852$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(75))$	10
Old forms	14
Cusp forms $S_{4}(\Gamma_0(75))$	24
Eisenstein series $E_{4}(\Gamma_0(75))$	12
Modular forms $M_{4}(\Gamma_0(75))$	36

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(75)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(25))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.111 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]: [1, 2, 3, 8, 9, 32] (certified at ell = 2147482921)
T = [(2, 1), (7, 3)]; h of degree 10 = 1 x 10 (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(75, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 75:4 1000

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