Atlas › Modular forms › Weight 2 › Level 760 › 760.2.a

Newspace 760.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(760))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
760.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + q^{5} + O(q^{8})$	− −	760d1 rank 1
760.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} + q^{5} + 4q^{7} + O(q^{8})$	− +	760b1 rank 0
760.2.a.c	1	$\mathbb{Q}$	$q + q^{5} + O(q^{8})$	− +	760e1 rank 1
760.2.a.d	1	$\mathbb{Q}$	$q + 2q^{3} + q^{5} + 4q^{7} + O(q^{8})$	− +	760a1 rank 0
760.2.a.e	1	$\mathbb{Q}$	$q + 3q^{3} + q^{5} - q^{7} + O(q^{8})$	− +	760c1 rank 0
760.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{5} + O(q^{8})$	+ +
760.2.a.g	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} - 2q^{5} + O(q^{8})$	+ −
760.2.a.h	3	degree 3	$3q - q^{3} - 3q^{5} - 5q^{7} + O(q^{8})$	+ +
760.2.a.i	3	degree 3	$3q - q^{3} - 3q^{5} - q^{7} + O(q^{8})$	+ −
760.2.a.j	3	degree 3	$3q - q^{3} + 3q^{5} - q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17} + 11T_{23}$ 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 $T$
760.2.a.a	$x + 56$
760.2.a.b	$x + 16$
760.2.a.c	$x + 28$
760.2.a.d	$x - 136$
760.2.a.e	$x + 91$
760.2.a.f	$x^{2} - 172x + 7268$
760.2.a.g	$x^{2} - 152x + 5476$
760.2.a.h	$x^{3} + 199x^{2} + 11200x + 128300$
760.2.a.i	$x^{3} + 207x^{2} + 11852x + 117932$
760.2.a.j	$x^{3} - 65x^{2} - 4384x + 84544$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(760))$	18
Old forms	95
Cusp forms $S_{2}(\Gamma_0(760))$	113
Eisenstein series $E_{2}(\Gamma_0(760))$	15
Modular forms $M_{2}(\Gamma_0(760))$	128

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.714 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, 4, 5, 8, 10, 19, 20, 38, 40, 76, 95, 152, 190, 380, 760]: [0, 1, 2, 1, 3, 3, 2, 6, 7, 10, 13, 12, 24, 34, 66, 128] (certified at ell = 2147483629)
T = [(3, 1), (7, 3), (11, 5), (13, 7), (17, 9), (23, 11)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 5 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(760, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 760:2 1000

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