Atlas › Modular forms › Weight 2 › Level 575 › 575.2.a › 575.2.a.l

Newform orbit 575.2.a.l

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + q^{2} + 11q^{4} + 5q^{6} - 3q^{7} + 6q^{8} + 15q^{9} - q^{11} - 6q^{12} + 3q^{13} + 7q^{14} + 7q^{16} - 10q^{17} + 24q^{18} + 15q^{19} + 2q^{21} - 21q^{22} + 7q^{23} + 18q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} - x^{6} - 35x^{5} + 11x^{4} + 303x^{3} - 74x^{2} - 658x - 7$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 575$ )
23	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	1	0	0	−3	−1	3	−10	15	7	3	14	10	19	−5	14	−4	−16	40	4	−14	3	−1	−17	16	24

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.318 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 5, 23, 25, 115, 575]: [0, 1, 3, 3, 14, 62] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 35 = 1 x 35 (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(575, 2)[11]   # 575.2.a.l

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

cargo run --release -p sagebrush-web --example atlas -- 575:2 1000