Atlas › Modular forms › Weight 2 › Level 754 › 754.2.a

Newspace 754.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(754))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
754.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} - 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	+ − −	$-1$	754c1 rank 1
754.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{7} + O(q^{8})$	+ − −	$-1$	754b1 rank 1
754.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 3q^{5} - q^{6} - q^{7} + O(q^{8})$	+ − +	$+1$	754a1 rank 0
754.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 3q^{5} + q^{6} - 3q^{7} + O(q^{8})$	− + −	$-1$	754d1 rank 1
754.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + q^{5} - 2q^{6} + 3q^{7} + O(q^{8})$	+ − +	$+1$
754.2.a.f	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	− − +	$-1$
754.2.a.g	4	degree 4	$4q - 4q^{2} + 4q^{4} + 7q^{5} + 3q^{7} + O(q^{8})$	+ + −	$+1$
754.2.a.h	4	degree 4	$4q - 4q^{2} + 2q^{3} + 4q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ + +	$-1$
754.2.a.i	5	degree 5	$5q + 5q^{2} + 3q^{3} + 5q^{4} + 6q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	− + +	$+1$
754.2.a.j	6	degree 6	$6q + 6q^{2} + 2q^{3} + 6q^{4} + 3q^{5} + 2q^{6} - q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{31}$ 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 $T$
754.2.a.a	$x + 58$
754.2.a.b	$x + 67$
754.2.a.c	$x - 77$
754.2.a.d	$x - 11$
754.2.a.e	$x^{2} - 119x + 3329$
754.2.a.f	$x^{2} + 174x + 5841$
754.2.a.g	$x^{4} - 107x^{3} + 4004x^{2} - 62615x + 345953$
754.2.a.h	$x^{4} + 278x^{3} + 21797x^{2} + 224740x - 20161444$
754.2.a.i	$x^{5} - 92x^{4} - 14340x^{3} + 1231921x^{2} + 14962028x - 572607676$
754.2.a.j	$x^{6} - 41x^{5} - 6536x^{4} + 88467x^{3} + 8168261x^{2} + 32619228x - 337574068$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(754))$	27
Old forms	74
Cusp forms $S_{2}(\Gamma_0(754))$	101
Eisenstein series $E_{2}(\Gamma_0(754))$	7
Modular forms $M_{2}(\Gamma_0(754))$	108

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(754)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(58))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(377))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.506 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, 13, 26, 29, 58, 377, 754]: [0, 1, 1, 5, 3, 9, 36, 108] (certified at ell = 2147483353)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (17, 5), (19, 6), (23, 7), (31, 8)]; h of degree 27 = 1 x 27 (phi(ord eps) x dim S_k^new); 8 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(754, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 754:2 1000

← 753.2.a · 755.2.a →