Atlas › Modular forms › Weight 2 › Level 428 › 428.2.a › 428.2.a.c

Newform orbit 428.2.a.c

Trace form

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

$2q - 3q^{3} - q^{5} - 2q^{7} + 5q^{9} + 2q^{11} - 4q^{13} - 5q^{15} - 9q^{17} - 6q^{19} + 3q^{21} - 8q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{13})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{2} + 6x - 4$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 428$ )
107	$-1$

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
$\operatorname{tr} a_p$	0	−3	−1	−2	2	−4	−9	−6	−8	10	−12	11	−2	−5	2	−2	−3	−1	−6	15	0	−7	11	0	−12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.203 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, 2, 4, 107, 214, 428]: [0, 1, 2, 10, 29, 57] (certified at ell = 2147482621)
T = [(3, 1), (5, 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(428, 2)[2]   # 428.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 428:2 1000