Atlas › Modular forms › Weight 4 › Level 245 › 245.4.a

Newspace 245.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(245))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
245.4.a.a	1	$\mathbb{Q}$	$q - 4q^{2} - 2q^{3} + 8q^{4} + 5q^{5} + 8q^{6} + O(q^{8})$	−
245.4.a.b	1	$\mathbb{Q}$	$q + q^{2} - 6q^{3} - 7q^{4} + 5q^{5} - 6q^{6} + O(q^{8})$	−
245.4.a.c	1	$\mathbb{Q}$	$q + q^{2} + 6q^{3} - 7q^{4} - 5q^{5} + 6q^{6} + O(q^{8})$	+
245.4.a.d	1	$\mathbb{Q}$	$q + q^{2} + 8q^{3} - 7q^{4} + 5q^{5} + 8q^{6} + O(q^{8})$	−
245.4.a.e	1	$\mathbb{Q}$	$q + 3q^{2} - 2q^{3} + q^{4} + 5q^{5} - 6q^{6} + O(q^{8})$	−
245.4.a.f	1	$\mathbb{Q}$	$q + 3q^{2} + 2q^{3} + q^{4} - 5q^{5} + 6q^{6} + O(q^{8})$	+
245.4.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q - 6q^{2} - 2q^{3} + 6q^{4} + 10q^{5} - 6q^{6} + O(q^{8})$	−
245.4.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q - 6q^{2} + 2q^{3} + 6q^{4} - 10q^{5} + 6q^{6} + O(q^{8})$	+
245.4.a.i	2	$\mathbb{Q}(\sqrt{11})$	$2q + 2q^{2} - 10q^{3} + 8q^{4} - 10q^{5} - 10q^{6} + O(q^{8})$	+
245.4.a.j	2	$\mathbb{Q}(\sqrt{11})$	$2q + 2q^{2} + 10q^{3} + 8q^{4} + 10q^{5} + 10q^{6} + O(q^{8})$	−
245.4.a.k	2	$\mathbb{Q}(\sqrt{2})$	$2q + 8q^{2} - 2q^{3} + 20q^{4} + 10q^{5} + 8q^{6} + O(q^{8})$	−
245.4.a.l	3	degree 3	$3q - 3q^{2} - 2q^{3} + 13q^{4} - 15q^{5} - 24q^{6} + O(q^{8})$	+
245.4.a.m	5	degree 5	$5q + q^{2} - 8q^{3} + 35q^{4} - 25q^{5} + 16q^{6} + O(q^{8})$	+
245.4.a.n	5	degree 5	$5q + q^{2} + 8q^{3} + 35q^{4} + 25q^{5} - 16q^{6} + O(q^{8})$	−
245.4.a.o	6	degree 6	$6q - 2q^{2} - 16q^{3} + 14q^{4} + 30q^{5} - 24q^{6} + O(q^{8})$	−
245.4.a.p	6	degree 6	$6q - 2q^{2} + 16q^{3} + 14q^{4} - 30q^{5} + 24q^{6} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
245.4.a.a	$x + 10$
245.4.a.b	$x + 17$
245.4.a.c	$x - 19$
245.4.a.d	$x - 25$
245.4.a.e	$x + 3$
245.4.a.f	$x - 9$
245.4.a.g	$x^{2} + 12x - 92$
245.4.a.h	$x^{2} - 200$
245.4.a.i	$x^{2} + 28x + 185$
245.4.a.j	$x^{2} - 32x + 245$
245.4.a.k	$x^{2} - 2x - 337$
245.4.a.l	$x^{3} + 9x^{2} - 635x + 585$
245.4.a.m	$x^{5} + 23x^{4} - 793x^{3} - 13059x^{2} + 173424x + 1236772$
245.4.a.n	$x^{5} - 25x^{4} - 649x^{3} + 12597x^{2} + 80340x - 652400$
245.4.a.o	$x^{6} + 50x^{5} + 247x^{4} - 14668x^{3} - 75889x^{2} + 1263354x - 2563807$
245.4.a.p	$x^{6} - 46x^{5} - 89x^{4} + 16556x^{3} + 37007x^{2} - 847710x + 536609$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(245))$	41
Old forms	35
Cusp forms $S_{4}(\Gamma_0(245))$	76
Eisenstein series $E_{4}(\Gamma_0(245))$	16
Modular forms $M_{4}(\Gamma_0(245))$	92

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(245)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.186 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, 5, 7, 35, 49, 245]: [1, 3, 3, 14, 15, 86] (certified at ell = 2147483353)
T = [(2, 1), (3, 3)]; h of degree 41 = 1 x 41 (phi(ord eps) x dim S_k^new); 8 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(245, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 245:4 1000

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