Atlas › Modular forms › Weight 4 › Level 242 › 242.4.a

Newspace 242.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(242))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
242.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 4q^{4} - 3q^{5} - 2q^{6} + 10q^{7} + O(q^{8})$	+
242.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 4q^{3} + 4q^{4} + 3q^{5} - 8q^{6} - 8q^{7} + O(q^{8})$	+
242.4.a.c	1	$\mathbb{Q}$	$q - 2q^{2} + 5q^{3} + 4q^{4} - 15q^{5} - 10q^{6} + 36q^{7} + O(q^{8})$	+
242.4.a.d	1	$\mathbb{Q}$	$q + 2q^{2} - 7q^{3} + 4q^{4} - 19q^{5} - 14q^{6} - 14q^{7} + O(q^{8})$	−
242.4.a.e	1	$\mathbb{Q}$	$q + 2q^{2} + 4q^{3} + 4q^{4} + 3q^{5} + 8q^{6} + 8q^{7} + O(q^{8})$	−
242.4.a.f	1	$\mathbb{Q}$	$q + 2q^{2} + 4q^{3} + 4q^{4} + 14q^{5} + 8q^{6} + 8q^{7} + O(q^{8})$	−
242.4.a.g	1	$\mathbb{Q}$	$q + 2q^{2} + 5q^{3} + 4q^{4} - 15q^{5} + 10q^{6} - 36q^{7} + O(q^{8})$	−
242.4.a.h	2	$\mathbb{Q}(\sqrt{5})$	$2q - 4q^{2} - 13q^{3} + 8q^{4} - q^{5} + 26q^{6} - 5q^{7} + O(q^{8})$	+
242.4.a.i	2	$\mathbb{Q}(\sqrt{3})$	$2q - 4q^{2} - 2q^{3} + 8q^{4} - 12q^{5} + 4q^{6} + 6q^{7} + O(q^{8})$	+
242.4.a.j	2	$\mathbb{Q}(\sqrt{37})$	$2q - 4q^{2} + 6q^{3} + 8q^{4} + 4q^{5} - 12q^{6} - 42q^{7} + O(q^{8})$	+
242.4.a.k	2	$\mathbb{Q}(\sqrt{5})$	$2q + 4q^{2} - 13q^{3} + 8q^{4} - q^{5} - 26q^{6} + 5q^{7} + O(q^{8})$	−
242.4.a.l	2	$\mathbb{Q}(\sqrt{3})$	$2q + 4q^{2} - 2q^{3} + 8q^{4} - 12q^{5} - 4q^{6} - 6q^{7} + O(q^{8})$	−
242.4.a.m	2	$\mathbb{Q}(\sqrt{37})$	$2q + 4q^{2} + 6q^{3} + 8q^{4} + 4q^{5} + 12q^{6} + 42q^{7} + O(q^{8})$	−
242.4.a.n	4	degree 4	$4q - 8q^{2} + 4q^{3} + 16q^{4} + 25q^{5} - 8q^{6} - 3q^{7} + O(q^{8})$	+
242.4.a.o	4	degree 4	$4q + 8q^{2} + 4q^{3} + 16q^{4} + 25q^{5} + 8q^{6} + 3q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
242.4.a.a	$x - 1466$
242.4.a.b	$x - 131$
242.4.a.c	$x - 590$
242.4.a.d	$x + 1158$
242.4.a.e	$x - 393$
242.4.a.f	$x + 1554$
242.4.a.g	$x - 830$
242.4.a.h	$x^{2} - 237x - 2193759$
242.4.a.i	$x^{2} - 3130x + 2421577$
242.4.a.j	$x^{2} + 2652x + 1592183$
242.4.a.k	$x^{2} + 1835x - 415705$
242.4.a.l	$x^{2} + 6290x + 9735073$
242.4.a.m	$x^{2} - 636x - 45729$
242.4.a.n	$x^{4} + 2437x^{3} - 3765534x^{2} - 9380166995x - 301182861445$
242.4.a.o	$x^{4} - 6941x^{3} + 13685068x^{2} - 6462941247x + 388868963139$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(242))$	27
Old forms	60
Cusp forms $S_{4}(\Gamma_0(242))$	87
Eisenstein series $E_{4}(\Gamma_0(242))$	24
Modular forms $M_{4}(\Gamma_0(242))$	111

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(242)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(22))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(121))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.55 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, 11, 22, 121, 242]: [1, 2, 4, 11, 34, 101] (certified at ell = 2147482921)
T = [(3, 1), (5, 2), (7, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 12 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(242, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 242:4 1000

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