Atlas › Modular forms › Weight 2 › Level 475 › 475.2.a › 475.2.a.f

Newform orbit 475.2.a.f

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{3} + 3x^{2} - x - 1$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 475$ )
19	$+1$

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	−2	0	0	−8	−8	−2	−3	4	−10	4	−20	−2	4	0	−16	−20	−2	−2	−4	−2	0	32	2	−20

$\operatorname{tr} a_n$ 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, 19, 25, 95, 475]: [0, 1, 2, 3, 12, 52] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 3 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(475, 2)[5]   # 475.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 475:2 1000