Atlas › Modular forms › Weight 2 › Level 762 › 762.2.a

Newspace 762.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(762))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
762.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 3q^{5} - q^{6} + 3q^{7} + O(q^{8})$	+ − −	$-1$	762c1 rank 1
762.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$	762a1 rank 0
762.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 3q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$	762b1 rank 0
762.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− + −	$-1$	762d1 rank 1
762.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 3q^{5} + q^{6} - 5q^{7} + O(q^{8})$	− − +	$-1$	762e1 rank 1
762.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + O(q^{8})$	− − −	$+1$	762g1 rank 0
762.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + O(q^{8})$	− − −	$+1$	762f1 rank 0
762.2.a.h	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + q^{5} + 2q^{6} + 6q^{7} + O(q^{8})$	+ + −	$+1$
762.2.a.i	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - q^{5} - 2q^{6} - 5q^{7} + O(q^{8})$	+ − −	$-1$
762.2.a.j	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 5q^{5} + 2q^{6} + 3q^{7} + O(q^{8})$	− − −	$+1$
762.2.a.k	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} + 4q^{5} + 3q^{6} - 6q^{7} + O(q^{8})$	+ + +	$-1$
762.2.a.l	5	degree 5	$5q + 5q^{2} - 5q^{3} + 5q^{4} + 2q^{5} - 5q^{6} + 5q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ 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 $T$
762.2.a.a	$x + 279$
762.2.a.b	$x - 158$
762.2.a.c	$x + 19$
762.2.a.d	$x + 115$
762.2.a.e	$x + 163$
762.2.a.f	$x - 129$
762.2.a.g	$x + 94$
762.2.a.h	$x^{2} - 215x - 4258$
762.2.a.i	$x^{2} + 162x + 2349$
762.2.a.j	$x^{2} + 118x + 2501$
762.2.a.k	$x^{3} + 495x^{2} + 70011x + 2725029$
762.2.a.l	$x^{5} - 573x^{4} + 99463x^{3} - 4615075x^{2} - 256797844x + 19702848524$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(762))$	21
Old forms	104
Cusp forms $S_{2}(\Gamma_0(762))$	125
Eisenstein series $E_{2}(\Gamma_0(762))$	7
Modular forms $M_{2}(\Gamma_0(762))$	132

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(762)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(127))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(254))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(381))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.108 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, 127, 254, 381, 762]: [0, 1, 1, 3, 11, 34, 44, 132] (certified at ell = 2147483647)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12)]; h of degree 21 = 1 x 21 (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(762, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 762:2 1000

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