Atlas › Modular forms › Weight 2 › Level 782 › 782.2.a

Newspace 782.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(782))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
782.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + 2q^{5} + 2q^{6} + O(q^{8})$	+ + +	$-1$	782a1 rank 1
782.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} + 4q^{7} + O(q^{8})$	− + +	$+1$	782c1 rank 0
782.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} + O(q^{8})$	− − −	$+1$	782e1 rank 0
782.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 4q^{5} + q^{6} + 3q^{7} + O(q^{8})$	− + +	$+1$	782b1 rank 0
782.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + 3q^{3} + q^{4} + 3q^{6} - q^{7} + O(q^{8})$	− + +	$+1$	782d1 rank 0
782.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{4} - 2q^{5} - 3q^{7} + O(q^{8})$	+ − −	$-1$
782.2.a.g	2	$\mathbb{Q}(\sqrt{21})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$
782.2.a.h	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} - 3q^{7} + O(q^{8})$	− − +	$-1$
782.2.a.i	2	$\mathbb{Q}(\sqrt{37})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 6q^{5} + 2q^{6} + 3q^{7} + O(q^{8})$	− + +	$+1$
782.2.a.j	3	degree 3	$3q + 3q^{2} - 2q^{3} + 3q^{4} - 4q^{5} - 2q^{6} - 7q^{7} + O(q^{8})$	− + −	$-1$
782.2.a.k	4	degree 4	$4q + 4q^{2} + 4q^{4} - 2q^{5} + 5q^{7} + O(q^{8})$	− − −	$+1$
782.2.a.l	5	degree 5	$5q - 5q^{2} + 5q^{4} + 9q^{7} + O(q^{8})$	+ + −	$+1$
782.2.a.m	6	degree 6	$6q - 6q^{2} + 6q^{4} - 2q^{5} + 9q^{7} + O(q^{8})$	+ − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{19} + 7T_{29} + 8T_{31}$ 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 $T$
782.2.a.a	$x - 12$
782.2.a.b	$x - 124$
782.2.a.c	$x - 80$
782.2.a.d	$x - 84$
782.2.a.e	$x - 14$
782.2.a.f	$x^{2} - 7x - 769$
782.2.a.g	$x^{2} + 153x + 5427$
782.2.a.h	$x^{2} + 133x + 1891$
782.2.a.i	$x^{2} + 83x - 1617$
782.2.a.j	$x^{3} + 157x^{2} + 2847x - 186052$
782.2.a.k	$x^{4} - 55x^{3} + 585x^{2} - 2250x + 2900$
782.2.a.l	$x^{5} - 123x^{4} - 4689x^{3} + 830440x^{2} + 2854622x - 1385177352$
782.2.a.m	$x^{6} - 35x^{5} - 22185x^{4} + 83448x^{3} + 133908538x^{2} + 3533503616x - 11744517408$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(782))$	31
Old forms	74
Cusp forms $S_{2}(\Gamma_0(782))$	105
Eisenstein series $E_{2}(\Gamma_0(782))$	7
Modular forms $M_{2}(\Gamma_0(782))$	112

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(782)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(46))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(391))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.777 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, 17, 23, 34, 46, 391, 782]: [0, 1, 2, 3, 6, 8, 38, 112] (certified at ell = 2147482481)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (19, 6), (29, 7), (31, 8)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 9 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(782, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 782:2 1000

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