Atlas › Modular forms › Weight 2 › Level 758 › 758.2.a

Newspace 758.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(758))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
758.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} - q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$	758a1 rank 1
758.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 3q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$	758b1 rank 0
758.2.a.c	5	degree 5	$5q - 5q^{2} - q^{3} + 5q^{4} - 3q^{5} + q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
758.2.a.d	5	degree 5	$5q + 5q^{2} - 7q^{3} + 5q^{4} - 3q^{5} - 7q^{6} - 8q^{7} + O(q^{8})$	− −	$-1$
758.2.a.e	10	degree 10	$10q - 10q^{2} + 6q^{3} + 10q^{4} + 5q^{5} - 6q^{6} + 6q^{7} + O(q^{8})$	+ −	$+1$
758.2.a.f	10	degree 10	$10q + 10q^{2} + 6q^{3} + 10q^{4} - 3q^{5} + 6q^{6} + 10q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
758.2.a.a	$x + 3$
758.2.a.b	$x - 5$
758.2.a.c	$x^{5} + 4x^{4} - 11x^{3} - 12x^{2} + 28x - 11$
758.2.a.d	$x^{5} + 10x^{4} + 33x^{3} + 36x^{2} - 4x - 17$
758.2.a.e	$x^{10} - 11x^{9} - 7x^{8} + 366x^{7} - 439x^{6} - 4039x^{5} + 6340x^{4} + 16611x^{3} - 22147x^{2} - 22443x + 10039$
758.2.a.f	$x^{10} - 3x^{9} - 37x^{8} + 138x^{7} + 239x^{6} - 1635x^{5} + 1998x^{4} + 605x^{3} - 2223x^{2} + 487x + 455$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(758))$	32
Old forms	62
Cusp forms $S_{2}(\Gamma_0(758))$	94
Eisenstein series $E_{2}(\Gamma_0(758))$	3
Modular forms $M_{2}(\Gamma_0(758))$	97

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(758)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(379))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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, 379, 758]: [0, 1, 32, 97] (certified at ell = 2147481883)
T = [(3, 1), (5, 1)]; h of degree 32 = 1 x 32 (phi(ord eps) x dim S_k^new); 4 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(758, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 758:2 1000

← 757.2.a · 759.2.a →