Atlas › Modular forms › Weight 4 › Level 249 › 249.4.a

Newspace 249.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(249))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
249.4.a.a	9	degree 9	$9q - 7q^{2} + 27q^{3} + 19q^{4} - 47q^{5} - 21q^{6} - 43q^{7} + O(q^{8})$	− +	$-1$
249.4.a.b	9	degree 9	$9q + 3q^{2} - 27q^{3} + 23q^{4} - q^{5} - 9q^{6} - 43q^{7} + O(q^{8})$	+ −	$-1$
249.4.a.c	12	degree 12	$12q - 3q^{2} - 36q^{3} + 59q^{4} - 11q^{5} + 9q^{6} + 41q^{7} + O(q^{8})$	+ +	$+1$
249.4.a.d	12	degree 12	$12q + 7q^{2} + 36q^{3} + 55q^{4} + 43q^{5} + 21q^{6} + 41q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
249.4.a.a	$x^{9} + 54x^{8} + 777x^{7} - 3971x^{6} - 177770x^{5} - 1265583x^{4} - 1436876x^{3} + 12288840x^{2} + 34631760x + 24074784$
249.4.a.b	$x^{9} - 2x^{8} - 497x^{7} - 241x^{6} + 55194x^{5} + 102681x^{4} - 1584424x^{3} - 4638440x^{2} + 1193968x + 121632$
249.4.a.c	$x^{12} + 14x^{11} - 1002x^{10} - 13239x^{9} + 339931x^{8} + 3950150x^{7} - 47252928x^{6} - 384977251x^{5} + 3168919168x^{4} + 9185779304x^{3} - 80181311120x^{2} + 98584386576x + 20397960000$
249.4.a.d	$x^{12} - 50x^{11} + 200x^{10} + 23519x^{9} - 288403x^{8} - 2866856x^{7} + 53090580x^{6} + 22306625x^{5} - 2730712876x^{4} + 8047983704x^{3} + 9816117824x^{2} - 25381033840x - 16369604160$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(249))$	42
Old forms	40
Cusp forms $S_{4}(\Gamma_0(249))$	82
Eisenstein series $E_{4}(\Gamma_0(249))$	4
Modular forms $M_{4}(\Gamma_0(249))$	86

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(249)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(83))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.159 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, 83, 249]: [1, 2, 22, 86] (certified at ell = 2147483323)
T = [(2, 1), (5, 1)]; h of degree 42 = 1 x 42 (phi(ord eps) x dim S_k^new); 7 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(249, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 249:4 1000

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