Atlas › Modular forms › Weight 2 › Level 752 › 752.2.a

Newspace 752.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(752))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
752.2.a.a	1	$\mathbb{Q}$	$q + O(q^{8})$	−	752a1 rank 1
752.2.a.b	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{3} - 5q^{7} + O(q^{8})$	−
752.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q + 4q^{5} + 4q^{7} + O(q^{8})$	+
752.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{3} - 4q^{5} - q^{7} + O(q^{8})$	+
752.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{3} - 2q^{5} + 3q^{7} + O(q^{8})$	−
752.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{3} - 2q^{5} + 7q^{7} + O(q^{8})$	+
752.2.a.g	4	degree 4	$4q - 3q^{3} - 5q^{7} + O(q^{8})$	+
752.2.a.h	4	degree 4	$4q - 2q^{5} - 4q^{7} + O(q^{8})$	+
752.2.a.i	4	degree 4	$4q + q^{3} + 4q^{5} + 3q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{13}$ 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 $T$
752.2.a.a	$x + 50$
752.2.a.b	$x^{2} + 4x - 1049$
752.2.a.c	$x^{2} + 60x + 882$
752.2.a.d	$x^{2} + 42x + 421$
752.2.a.e	$x^{2} - 70x + 725$
752.2.a.f	$x^{2} - 24x + 99$
752.2.a.g	$x^{4} + 174x^{3} + 8271x^{2} + 103468x + 383510$
752.2.a.h	$x^{4} - 88x^{3} - 3582x^{2} + 333768x - 3452483$
752.2.a.i	$x^{4} - 108x^{3} + 3969x^{2} - 57682x + 264716$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(752))$	23
Old forms	68
Cusp forms $S_{2}(\Gamma_0(752))$	91
Eisenstein series $E_{2}(\Gamma_0(752))$	11
Modular forms $M_{2}(\Gamma_0(752))$	102

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(752)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(94))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(188))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(376))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.345 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, 8, 16, 47, 94, 188, 376, 752]: [0, 1, 2, 3, 4, 5, 14, 27, 52, 100] (certified at ell = 2147483137)
T = [(3, 1), (5, 3), (7, 5), (11, 7), (13, 9)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 6 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(752, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 752:2 1000

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