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

Newform orbit 175.4.a.g

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - 4q^{2} + 3q^{3} + 36q^{4} + q^{6} - 28q^{7} - 27q^{8} + 61q^{9} + 100q^{11} + 165q^{12} - 44q^{13} + 28q^{14} + 160q^{16} + 53q^{17} - 433q^{18} - 29q^{19} - 21q^{21} + 152q^{22} - 295q^{23} - 21q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{4} - 5x^{3} - 785x^{2} + 2872x + 121728$

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
$\operatorname{tr} a_p$	−4	3	0	−28	100	−44	53	−29	−295	129	114	−403	671	411	8	−90	1018	50	−424	215	1207	−951	3035	2819	1100

$\operatorname{tr} a_n$ 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)[6]   # 175.4.a.g

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

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