Atlas › Modular forms › Weight 2 › Level 790 › 790.2.a

Newspace 790.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(790))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
790.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} + q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− − +	$-1$	790a1 rank 1
790.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} + 2q^{5} + O(q^{8})$	+ − −	$-1$
790.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ + −	$+1$
790.2.a.d	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{2} + 2q^{4} - 2q^{5} - 6q^{7} + O(q^{8})$	− + −	$-1$
790.2.a.e	4	degree 4	$4q - 4q^{2} - 2q^{3} + 4q^{4} - 4q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	+ + +	$-1$
790.2.a.f	4	degree 4	$4q - 4q^{2} - 2q^{3} + 4q^{4} + 4q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	+ − +	$+1$
790.2.a.g	4	degree 4	$4q + 4q^{2} + 2q^{3} + 4q^{4} - 4q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	− + +	$+1$
790.2.a.h	6	degree 6	$6q + 6q^{2} + 2q^{3} + 6q^{4} + 6q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7}$ 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 $T$
790.2.a.a	$x + 8$
790.2.a.b	$x^{2} - 50$
790.2.a.c	$x^{2} + 4x - 44$
790.2.a.d	$x^{2} + 18x + 54$
790.2.a.e	$x^{4} + 8x^{3} - 54x^{2} - 548x - 860$
790.2.a.f	$x^{4} + 14x^{3} - 62x^{2} - 480x + 52$
790.2.a.g	$x^{4} - 14x^{3} - 36x^{2} + 92x + 76$
790.2.a.h	$x^{6} - 14x^{5} - 120x^{4} + 2136x^{3} - 1748x^{2} - 36184x + 1032$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(790))$	25
Old forms	92
Cusp forms $S_{2}(\Gamma_0(790))$	117
Eisenstein series $E_{2}(\Gamma_0(790))$	7
Modular forms $M_{2}(\Gamma_0(790))$	124

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(790)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(79))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(158))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(395))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.296 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, 5, 10, 79, 158, 395, 790]: [0, 1, 1, 3, 7, 22, 42, 124] (certified at ell = 2147483053)
T = [(3, 1), (7, 3)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 4 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(790, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 790:2 1000

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