Atlas › Modular forms › Weight 4 › Level 248 › 248.4.a

Newspace 248.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(248))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
248.4.a.a	4	degree 4	$4q - 2q^{3} + 9q^{5} - 45q^{7} + O(q^{8})$	−
248.4.a.b	5	degree 5	$5q - 6q^{3} - 17q^{5} + 3q^{7} + O(q^{8})$	+
248.4.a.c	6	degree 6	$6q + 10q^{3} - 11q^{5} + 11q^{7} + O(q^{8})$	+
248.4.a.d	7	degree 7	$7q + 23q^{5} + 31q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
248.4.a.a	$x^{4} - 7x^{3} - 155x^{2} - 57x + 2322$
248.4.a.b	$x^{5} + 23x^{4} + 19x^{3} - 1921x^{2} - 6500x + 19394$
248.4.a.c	$x^{6} + x^{5} - 831x^{4} + 1549x^{3} + 157906x^{2} - 72798x - 7508988$
248.4.a.d	$x^{7} - 23x^{6} - 513x^{5} + 14157x^{4} + 27152x^{3} - 1819046x^{2} + 1155920x + 63534200$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(248))$	22
Old forms	70
Cusp forms $S_{4}(\Gamma_0(248))$	92
Eisenstein series $E_{4}(\Gamma_0(248))$	8
Modular forms $M_{4}(\Gamma_0(248))$	100

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(248)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(8))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(62))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(124))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.156 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, 31, 62, 124, 248]: [1, 2, 3, 5, 9, 26, 51, 100] (certified at ell = 2147482951)
T = [(3, 1), (5, 1)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 4 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(248, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 248:4 1000

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