Atlas › Modular forms › Weight 4 › Level 175 › 175.4.a › 175.4.a.e

Newform orbit 175.4.a.e

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + q^{2} - 5q^{3} + 5q^{4} - 23q^{6} + 14q^{7} + 15q^{8} - 21q^{9} - 56q^{11} - 33q^{12} + 52q^{13} + 7q^{14} - 135q^{16} - 103q^{17} + 92q^{18} - 57q^{19} - 35q^{21} - 233q^{22} - 31q^{23} + 65q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{41})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} + 14x + 8$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 175$ )
7	$-1$

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	−5	0	14	−56	52	−103	−57	−31	−413	−162	75	−505	−73	224	−262	190	990	908	127	−337	−1119	−1517	−1713	1764

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.139 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, 5, 7, 25, 35, 175]: [1, 3, 3, 9, 14, 62] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 29 = 1 x 29 (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(175, 4)[4]   # 175.4.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 175:4 1000