Atlas › Modular forms › Weight 2 › Level 793 › 793.2.a

Newspace 793.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(793))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
793.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + 2q^{5} - 4q^{7} + O(q^{8})$	+ +	$-1$	793a1 rank 1
793.2.a.b	4	degree 4	$4q - q^{2} - 3q^{3} + 3q^{4} - 7q^{5} + 5q^{6} - q^{7} + O(q^{8})$	+ +	$-1$
793.2.a.c	9	degree 9	$9q - 3q^{2} - 7q^{3} + 9q^{4} + q^{5} - 5q^{6} - 7q^{7} + O(q^{8})$	+ +	$-1$
793.2.a.d	15	degree 15	$15q - 8q^{2} - 6q^{3} + 18q^{4} - 8q^{5} - 6q^{6} - 12q^{7} + O(q^{8})$	− −	$-1$
793.2.a.e	16	degree 16	$16q + 3q^{2} + 6q^{3} + 17q^{4} + 4q^{5} + 8q^{6} + 10q^{7} + O(q^{8})$	+ −	$+1$
793.2.a.f	16	degree 16	$16q + 5q^{2} + 6q^{3} + 17q^{4} + 10q^{5} + 2q^{6} + 10q^{7} + O(q^{8})$	− +	$+1$

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 $T$
793.2.a.a	$x - 1$
793.2.a.b	$x^{4} + 4x^{3} - 9x^{2} - 26x + 39$
793.2.a.c	$x^{9} + 10x^{8} + 22x^{7} - 63x^{6} - 316x^{5} - 431x^{4} - 246x^{3} - 54x^{2} + 1$
793.2.a.d	$x^{15} + 14x^{14} + 50x^{13} - 105x^{12} - 872x^{11} - 493x^{10} + 4525x^{9} + 6356x^{8} - 7628x^{7} - 18048x^{6} - 2348x^{5} + 14103x^{4} + 10392x^{3} + 2388x^{2} + 94x + 1$
793.2.a.e	$x^{16} - 9x^{15} - 26x^{14} + 417x^{13} - 249x^{12} - 6189x^{11} + 8302x^{10} + 46039x^{9} - 66712x^{8} - 198554x^{7} + 235140x^{6} + 502351x^{5} - 345777x^{4} - 638838x^{3} + 111306x^{2} + 207797x - 20741$
793.2.a.f	$x^{16} - 11x^{15} + 4x^{14} + 331x^{13} - 895x^{12} - 2843x^{11} + 13302x^{10} + 375x^{9} - 64116x^{8} + 74514x^{7} + 60360x^{6} - 164495x^{5} + 94645x^{4} + 1080x^{3} - 14608x^{2} + 2041x + 413$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(793))$	61
Old forms	8
Cusp forms $S_{2}(\Gamma_0(793))$	69
Eisenstein series $E_{2}(\Gamma_0(793))$	3
Modular forms $M_{2}(\Gamma_0(793))$	72

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(793)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(61))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.243 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, 61, 793]: [0, 1, 5, 72] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 61 = 1 x 61 (phi(ord eps) x dim S_k^new); 6 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(793, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 793:2 1000

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