Atlas › Modular forms › Weight 2 › Level 913 › 913.2.a › 913.2.a.g

Newform orbit 913.2.a.g

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q + 9q^{2} + 5q^{3} + 17q^{4} + 2q^{5} + 3q^{6} + 13q^{7} + 15q^{8} + 16q^{9} - 3q^{10} + 17q^{11} - q^{12} + 11q^{13} + 10q^{14} + 7q^{15} + 17q^{16} + 14q^{17} + 9q^{18} + 8q^{19} - 10q^{20} + q^{21} + 9q^{22} + 37q^{23} + 29q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{17} - 14x^{16} + 36x^{15} + 341x^{14} - 2099x^{13} + 962x^{12} + 19245x^{11} - 45968x^{10} - 4288x^{9} + 118782x^{8} - 94248x^{7} - 59487x^{6} + 92762x^{5} - 14625x^{4} - 18074x^{3} + 7390x^{2} - 725x + 21$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$
83	$+1$

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
$\operatorname{tr} a_p$	9	5	2	13	17	11	14	8	37	27	−6	−12	0	29	17	24	−12	−2	0	38	−4	27	−17	−1	−17

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.268 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, 11, 83, 913]: [0, 2, 8, 86] (certified at ell = 2147478731)
T = [(2, 1), (3, 1)]; h of degree 67 = 1 x 67 (phi(ord eps) x dim S_k^new); 7 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(913, 2)[6]   # 913.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 913:2 1000