Atlas › Modular forms › Weight 2 › Level 403 › 403.2.a › 403.2.a.e

Newform orbit 403.2.a.e

Trace form

The 8 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$8q - q^{2} + 7q^{3} + 7q^{4} + 11q^{5} - 2q^{7} + 9q^{9} - 2q^{10} - 2q^{11} + 19q^{12} + 8q^{13} + 3q^{14} + 6q^{15} - 7q^{16} + 7q^{17} - 12q^{18} - 5q^{19} + 6q^{20} + 8q^{21} - 4q^{22} + 14q^{23} - 17q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 8. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{8} - 20x^{7} + 40x^{6} + 1463x^{5} - 9098x^{4} - 11319x^{3} + 206232x^{2} - 454413x + 215864$

Atkin–Lehner signs

$p$	$w_p$
13	$-1$
31	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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	7	11	−2	−2	8	7	−5	14	12	−8	2	13	−5	23	25	−5	−9	22	−17	−27	−23	−25	2	−15

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.191 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, 13, 31, 403]: [0, 1, 3, 38] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 5 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(403, 2)[4]   # 403.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 403:2 1000