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

Newform orbit 913.2.a.f

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$16q - 5q^{2} - 8q^{3} + 13q^{4} - 7q^{5} - 5q^{6} - 5q^{7} - 15q^{8} + 12q^{9} + q^{10} - 16q^{11} - 15q^{12} + 5q^{13} - 16q^{14} - 18q^{15} + 7q^{16} - 20q^{17} + 5q^{18} - 4q^{19} - 10q^{20} + 5q^{21} + 5q^{22} - 44q^{23} + 5q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} + 13x^{15} + 37x^{14} - 170x^{13} - 975x^{12} + 21x^{11} + 7204x^{10} + 8056x^{9} - 19536x^{8} - 37810x^{7} + 9008x^{6} + 53291x^{5} + 22733x^{4} - 12618x^{3} - 7232x^{2} + 804x + 369$

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 odd 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
$\operatorname{tr} a_p$	−5	−8	−7	−5	−16	5	−20	−4	−44	−7	−11	−15	0	−11	−31	−48	1	10	−33	−69	0	−19	−16	−30	−20

$\operatorname{tr} a_n$ 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)[5]   # 913.2.a.f

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

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