Atlas › Modular forms › Weight 4 › Level 246 › 246.4.a

Newspace 246.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(246))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
246.4.a.a	1	$\mathbb{Q}$	$q + 2q^{2} + 3q^{3} + 4q^{4} - 14q^{5} + 6q^{6} - 28q^{7} + O(q^{8})$	− − −	$-1$
246.4.a.b	2	$\mathbb{Q}(\sqrt{86})$	$2q - 4q^{2} - 6q^{3} + 8q^{4} - 24q^{5} + 12q^{6} + 32q^{7} + O(q^{8})$	+ + −	$-1$
246.4.a.c	2	$\mathbb{Q}(\sqrt{57})$	$2q - 4q^{2} - 6q^{3} + 8q^{4} + q^{5} + 12q^{6} + 4q^{7} + O(q^{8})$	+ + +	$+1$
246.4.a.d	2	$\mathbb{Q}(\sqrt{7})$	$2q - 4q^{2} + 6q^{3} + 8q^{4} - 2q^{5} - 12q^{6} - 14q^{7} + O(q^{8})$	+ − +	$-1$
246.4.a.e	2	$\mathbb{Q}(\sqrt{65})$	$2q + 4q^{2} - 6q^{3} + 8q^{4} - 6q^{5} - 12q^{6} - 16q^{7} + O(q^{8})$	− + +	$-1$
246.4.a.f	3	degree 3	$3q + 6q^{2} - 9q^{3} + 12q^{4} - q^{5} - 18q^{6} - 2q^{7} + O(q^{8})$	− + −	$+1$
246.4.a.g	4	degree 4	$4q - 8q^{2} + 12q^{3} + 16q^{4} + 3q^{5} - 24q^{6} + 14q^{7} + O(q^{8})$	+ − −	$+1$
246.4.a.h	4	degree 4	$4q + 8q^{2} + 12q^{3} + 16q^{4} + 11q^{5} + 24q^{6} + 42q^{7} + O(q^{8})$	− − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
246.4.a.a	$x + 98$
246.4.a.b	$x^{2} - 72x + 952$
246.4.a.c	$x^{2} - 13x - 10346$
246.4.a.d	$x^{2} + 44x + 232$
246.4.a.e	$x^{2} + 54x - 896$
246.4.a.f	$x^{3} + 7x^{2} - 5320x + 136904$
246.4.a.g	$x^{4} - 45x^{3} - 10634x^{2} + 283140x + 27995832$
246.4.a.h	$x^{4} - 137x^{3} + 1156x^{2} + 238168x - 5212224$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(246))$	20
Old forms	102
Cusp forms $S_{4}(\Gamma_0(246))$	122
Eisenstein series $E_{4}(\Gamma_0(246))$	8
Modular forms $M_{4}(\Gamma_0(246))$	130

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(246)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(41))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(82))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(123))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.227 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, 3, 6, 41, 82, 123, 246]: [1, 2, 2, 5, 12, 34, 44, 130] (certified at ell = 2147482921)
T = [(5, 1), (7, 3)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 5 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(246, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 246:4 1000

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