Atlas › Modular forms › Weight 2 › Level 748 › 748.2.a

Newspace 748.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(748))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
748.2.a.a	1	$\mathbb{Q}$	$q + 3q^{3} + 3q^{5} - 2q^{7} + O(q^{8})$	+ +	748a1 rank 0
748.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} + 2q^{5} - 4q^{7} + O(q^{8})$	+ −
748.2.a.c	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{3} - q^{5} + 5q^{7} + O(q^{8})$	+ +
748.2.a.d	3	degree 3	$3q - 3q^{3} - q^{5} - 4q^{7} + O(q^{8})$	− +
748.2.a.e	4	degree 4	$4q + q^{3} - q^{5} + q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{13} + 9T_{19}$ 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 $T$
748.2.a.a	$x - 6$
748.2.a.b	$x^{2} + 88x + 1904$
748.2.a.c	$x^{2} + 101x + 302$
748.2.a.d	$x^{3} + 76x^{2} - 2312x - 122572$
748.2.a.e	$x^{4} - 107x^{3} + 2296x^{2} - 12284x + 19252$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(748))$	12
Old forms	91
Cusp forms $S_{2}(\Gamma_0(748))$	103
Eisenstein series $E_{2}(\Gamma_0(748))$	11
Modular forms $M_{2}(\Gamma_0(748))$	114

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(748)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(44))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(68))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(187))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(374))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.328 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, 11, 17, 22, 34, 44, 68, 187, 374, 748]: [0, 1, 2, 2, 2, 5, 6, 9, 12, 20, 58, 114] (certified at ell = 2147482801)
T = [(3, 1), (5, 3), (7, 5), (13, 7), (19, 9)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 3 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(748, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 748:2 1000

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