Atlas › Modular forms › Weight 2 › Level 175 › 175.2.a › 175.2.a.f

Newform orbit 175.2.a.f

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{17})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{2} - 4x - 13$

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	1	0	2	1	−5	5	−6	2	1	0	−12	2	−10	5	2	−8	6	−4	16	8	−9	−8	6	9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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, 7, 25, 35, 175]: [0, 1, 1, 3, 6, 22] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 2 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, 2)[5]   # 175.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 175:2 1000