Atlas › Modular forms › Weight 2 › Level 513 › 513.2.a › 513.2.a.i

Newform orbit 513.2.a.i

Trace form

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

$4q + 8q^{4} + 12q^{7} + 4q^{10} - 4q^{16} - 4q^{19} - 12q^{22} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{4} - 60x^{3} - 848x^{2} + 52680x + 453028$

Atkin–Lehner signs

$p$	$w_p$
3	— ( $3^2 \mid 513$ )
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
$\operatorname{tr} a_p$	0	0	0	12	0	0	0	−4	0	0	8	−4	0	−4	0	0	0	−8	−16	0	16	−36	0	0	4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.342 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, 3, 9, 19, 27, 57, 171, 513]: [0, 1, 2, 2, 4, 8, 22, 62] (certified at ell = 2147483647)
T = [(2, 1), (5, 3), (7, 5), (11, 7)]; h of degree 24 = 1 x 24 (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(513, 2)[8]   # 513.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 513:2 1000