Atlas › Modular forms › Weight 4 › Level 221 › 221.4.a

Newspace 221.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(221))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
221.4.a.a	1	$\mathbb{Q}$	$q + 5q^{2} + 4q^{3} + 17q^{4} + 6q^{5} + 20q^{6} - 20q^{7} + O(q^{8})$	+ +	$+1$
221.4.a.b	10	degree 10	$10q - 3q^{2} - 4q^{3} + q^{4} - 12q^{5} - 40q^{7} + O(q^{8})$	− +	$-1$
221.4.a.c	10	degree 10	$10q + q^{2} - 16q^{3} + 37q^{4} - 6q^{5} - 40q^{6} - 18q^{7} + O(q^{8})$	+ −	$-1$
221.4.a.d	13	degree 13	$13q - 4q^{2} + 4q^{3} + 68q^{4} - 2q^{5} + 98q^{6} + 36q^{7} + O(q^{8})$	+ +	$+1$
221.4.a.e	14	degree 14	$14q + 5q^{2} + 20q^{3} + 49q^{4} + 18q^{5} - 38q^{6} + 94q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
221.4.a.a	$x - 9$
221.4.a.b	$x^{10} + 7x^{9} - 163x^{8} - 808x^{7} + 8846x^{6} + 23378x^{5} - 197146x^{4} - 104568x^{3} + 1344417x^{2} - 805049x - 914611$
221.4.a.c	$x^{10} + 15x^{9} - 53x^{8} - 1456x^{7} - 1274x^{6} + 37230x^{5} + 37994x^{4} - 345288x^{3} - 5847x^{2} + 668155x - 107365$
221.4.a.d	$x^{13} - 447x^{11} - 503x^{10} + 65567x^{9} + 162129x^{8} - 3853597x^{7} - 13893445x^{6} + 79805668x^{5} + 327643179x^{4} - 629231976x^{3} - 2658459048x^{2} + 1682976960x + 5860946664$
221.4.a.e	$x^{14} - 25x^{13} + 31x^{12} + 3232x^{11} - 14654x^{10} - 154630x^{9} + 795858x^{8} + 3811992x^{7} - 16854655x^{6} - 56192157x^{5} + 150450235x^{4} + 478314968x^{3} - 363820404x^{2} - 1756237236x - 1159497628$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(221))$	48
Old forms	14
Cusp forms $S_{4}(\Gamma_0(221))$	62
Eisenstein series $E_{4}(\Gamma_0(221))$	4
Modular forms $M_{4}(\Gamma_0(221))$	66

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(221)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.143 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, 13, 17, 221]: [1, 5, 6, 66] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 48 = 1 x 48 (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(221, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 221:4 1000

← 220.4.a · 222.4.a →