Atlas › Modular forms › Weight 2 › Level 792 › 792.2.a

Newspace 792.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(792))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
792.2.a.a	1	$\mathbb{Q}$	$q - 4q^{5} - 2q^{7} + O(q^{8})$	−	792g1 rank 0
792.2.a.b	1	$\mathbb{Q}$	$q - 2q^{5} + O(q^{8})$	+	792d1 rank 1
792.2.a.c	1	$\mathbb{Q}$	$q - 2q^{7} + O(q^{8})$	+	792a1 rank 1
792.2.a.d	1	$\mathbb{Q}$	$q - 2q^{7} + O(q^{8})$	−	792c1 rank 1
792.2.a.e	1	$\mathbb{Q}$	$q + 2q^{7} + O(q^{8})$	+	792b1 rank 0
792.2.a.f	1	$\mathbb{Q}$	$q + 2q^{5} + 4q^{7} + O(q^{8})$	−	792e1 rank 0
792.2.a.g	1	$\mathbb{Q}$	$q + 3q^{5} - 2q^{7} + O(q^{8})$	−	792f1 rank 0
792.2.a.h	2	$\mathbb{Q}(\sqrt{17})$	$2q - 3q^{5} - 2q^{7} + O(q^{8})$	−
792.2.a.i	2	$\mathbb{Q}(\sqrt{2})$	$2q + 4q^{7} + O(q^{8})$	+
792.2.a.j	2	$\mathbb{Q}(\sqrt{2})$	$2q + 4q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{13} + 4T_{19} + 5T_{23} + 6T_{29} + 7T_{17} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47} + 13T_{53} + 14T_{59} + 15T_{61} + 16T_{67}$ 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 $T$
792.2.a.a	$x - 174$
792.2.a.b	$x - 80$
792.2.a.c	$x + 238$
792.2.a.d	$x + 298$
792.2.a.e	$x - 522$
792.2.a.f	$x + 300$
792.2.a.g	$x - 125$
792.2.a.h	$x^{2} + 463x + 25708$
792.2.a.i	$x^{2} - 516x + 58876$
792.2.a.j	$x^{2} - 380x - 33092$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(792))$	13
Old forms	116
Cusp forms $S_{2}(\Gamma_0(792))$	129
Eisenstein series $E_{2}(\Gamma_0(792))$	31
Modular forms $M_{2}(\Gamma_0(792))$	160

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.739 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, 4, 6, 8, 9, 11, 12, 18, 22, 24, 33, 36, 44, 66, 72, 88, 99, 132, 198, 264, 396, 792]: [0, 1, 1, 2, 3, 3, 2, 2, 5, 5, 5, 8, 6, 9, 9, 16, 16, 16, 14, 30, 40, 56, 78, 152] (certified at ell = 2147482951)
T = [(5, 1), (7, 2), (13, 3), (19, 4), (23, 5), (29, 6), (17, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12), (53, 13), (59, 14), (61, 15), (67, 16)]; h of degree 13 = 1 x 13 (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(792, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 792:2 1000

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