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

Newform orbit 513.2.a.f

Trace form

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

$3q + q^{2} + 3q^{4} + 5q^{5} + q^{7} + 6q^{8} - 7q^{10} + 8q^{11} - q^{13} + 2q^{14} + 3q^{16} + 7q^{17} - 3q^{19} - 2q^{20} + q^{22} + 9q^{23} + O(q^{25})$

Coefficient field

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

$x^{3} - 77x^{2} + 1927x - 15747$

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
$\operatorname{tr} a_p$	1	0	5	1	8	−1	7	−3	9	−6	−2	−11	5	9	27	2	−1	7	−12	0	−3	7	5	−4	−23

$\operatorname{tr} a_n$ 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)[5]   # 513.2.a.f

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

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