Atlas › Modular forms › Weight 2 › Level 786 › 786.2.a

Newspace 786.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(786))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
786.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} + 2q^{7} + O(q^{8})$	+ + +	$-1$	786b1 rank 1
786.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 3q^{7} + O(q^{8})$	+ + −	$+1$	786d1 rank 0
786.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{7} + O(q^{8})$	+ + +	$-1$	786a1 rank 1
786.2.a.d	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} + O(q^{8})$	+ + −	$+1$	786e1 rank 0
786.2.a.e	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 4q^{5} + q^{6} - q^{7} + O(q^{8})$	+ + +	$-1$	786c1 rank 1
786.2.a.f	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 3q^{5} - q^{6} + 5q^{7} + O(q^{8})$	+ − +	$+1$	786f1 rank 0
786.2.a.g	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ − −	$-1$	786h1 rank 1
786.2.a.h	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - 5q^{7} + O(q^{8})$	+ − −	$-1$	786g1 rank 1
786.2.a.i	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 4q^{5} - q^{6} + 3q^{7} + O(q^{8})$	− + −	$-1$	786k1 rank 1
786.2.a.j	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− + −	$-1$	786j1 rank 1
786.2.a.k	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + 4q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− + +	$+1$	786i1 rank 0
786.2.a.l	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 3q^{5} + q^{6} - 3q^{7} + O(q^{8})$	− − +	$-1$	786l1 rank 1
786.2.a.m	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + 3q^{7} + O(q^{8})$	− − −	$+1$	786m1 rank 0
786.2.a.n	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} + 2q^{5} - 3q^{6} + 3q^{7} + O(q^{8})$	+ − +	$+1$
786.2.a.o	3	degree 3	$3q + 3q^{2} - 3q^{3} + 3q^{4} + q^{5} - 3q^{6} + q^{7} + O(q^{8})$	− + +	$+1$
786.2.a.p	4	degree 4	$4q + 4q^{2} + 4q^{3} + 4q^{4} + 4q^{6} + q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{17} + 4T_{19} + 5T_{23} + 6T_{11} + 7T_{13} + 8T_{29}$ 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 $T$
786.2.a.a	$x - 86$
786.2.a.b	$x - 33$
786.2.a.c	$x + 71$
786.2.a.d	$x - 54$
786.2.a.e	$x + 136$
786.2.a.f	$x - 47$
786.2.a.g	$x + 66$
786.2.a.h	$x + 21$
786.2.a.i	$x + 96$
786.2.a.j	$x + 111$
786.2.a.k	$x - 20$
786.2.a.l	$x - 15$
786.2.a.m	$x + 9$
786.2.a.n	$x^{3} - 74x^{2} - 1974x - 9136$
786.2.a.o	$x^{3} - 141x^{2} + 768x + 340624$
786.2.a.p	$x^{4} - 70x^{3} - 1482x^{2} + 56136x + 843776$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(786))$	23
Old forms	106
Cusp forms $S_{2}(\Gamma_0(786))$	129
Eisenstein series $E_{2}(\Gamma_0(786))$	7
Modular forms $M_{2}(\Gamma_0(786))$	136

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(786)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(131))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(262))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(393))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.886 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, 131, 262, 393, 786]: [0, 1, 1, 3, 12, 35, 46, 136] (certified at ell = 2147482481)
T = [(5, 1), (7, 2), (17, 3), (19, 4), (23, 5), (11, 6), (13, 7), (29, 8)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 7 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(786, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 786:2 1000

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