Atlas › Modular forms › Weight 4 › Level 250 › 250.4.a

Newspace 250.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(250))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
250.4.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 4q^{2} + q^{3} + 8q^{4} - 2q^{6} + 28q^{7} + O(q^{8})$	+
250.4.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 4q^{2} + 11q^{3} + 8q^{4} - 22q^{6} + 3q^{7} + O(q^{8})$	+
250.4.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q + 4q^{2} - 11q^{3} + 8q^{4} - 22q^{6} - 3q^{7} + O(q^{8})$	−
250.4.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + 4q^{2} - q^{3} + 8q^{4} - 2q^{6} - 28q^{7} + O(q^{8})$	−
250.4.a.e	4	degree 4	$4q - 8q^{2} - 8q^{3} + 16q^{4} + 16q^{6} - 34q^{7} + O(q^{8})$	+
250.4.a.f	4	degree 4	$4q - 8q^{2} - 3q^{3} + 16q^{4} + 6q^{6} - 9q^{7} + O(q^{8})$	+
250.4.a.g	4	degree 4	$4q + 8q^{2} + 3q^{3} + 16q^{4} + 6q^{6} + 9q^{7} + O(q^{8})$	−
250.4.a.h	4	degree 4	$4q + 8q^{2} + 8q^{3} + 16q^{4} + 16q^{6} + 34q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{7}$ 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 $T$
250.4.a.a	$x^{2} - 29x + 209$
250.4.a.b	$x^{2} - 14x - 31$
250.4.a.c	$x^{2} + 14x - 31$
250.4.a.d	$x^{2} + 29x + 209$
250.4.a.e	$x^{4} + 42x^{3} + 414x^{2} + 858x - 1199$
250.4.a.f	$x^{4} + 12x^{3} - 1261x^{2} - 27582x - 107129$
250.4.a.g	$x^{4} - 12x^{3} - 1261x^{2} + 27582x - 107129$
250.4.a.h	$x^{4} - 42x^{3} + 414x^{2} - 858x - 1199$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(250))$	24
Old forms	79
Cusp forms $S_{4}(\Gamma_0(250))$	103
Eisenstein series $E_{4}(\Gamma_0(250))$	20
Modular forms $M_{4}(\Gamma_0(250))$	123

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(250)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 6}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(25))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(50))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(125))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.19 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, 5, 10, 25, 50, 125, 250]: [1, 2, 3, 7, 9, 25, 39, 115] (certified at ell = 2147482801)
T = [(3, 1), (7, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 5 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(250, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 250:4 1000

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