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

Newform orbit 175.4.a.c

Trace form

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

$2q - 8q^{2} - 2q^{3} + 20q^{4} - 8q^{6} + 14q^{7} - 48q^{8} + 12q^{9} - 14q^{11} + 108q^{12} - 50q^{13} - 56q^{14} + 168q^{16} + 50q^{17} - 16q^{18} + 36q^{19} - 14q^{21} + 184q^{22} - 244q^{23} - 496q^{24} + O(q^{25})$

Coefficient field

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

$x^{2} + 14x - 193$

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
$\operatorname{tr} a_p$	−8	−2	0	14	−14	−50	50	36	−244	−26	−120	−564	−328	260	350	56	−1232	336	152	−1904	−676	1014	376	−216	−2742

$\operatorname{tr} a_n$ 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)[2]   # 175.4.a.c

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

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