Atlas › Modular forms › Weight 2 › Level 691 › 691.2.a › 691.2.a.a

Newform orbit 691.2.a.a

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q - 8q^{2} - 5q^{3} + 18q^{4} - 23q^{5} - 11q^{6} - 6q^{7} - 24q^{8} + 9q^{9} - 30q^{11} - 6q^{12} - 12q^{13} - 21q^{14} - 11q^{15} + 14q^{16} - 18q^{17} - 9q^{18} - 15q^{19} - 32q^{20} - 47q^{21} - 19q^{23} - 21q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{24} + 13x^{23} + 22x^{22} - 365x^{21} - 1475x^{20} + 3076x^{19} + 21492x^{18} - 6897x^{17} - 152462x^{16} - 33583x^{15} + 630495x^{14} + 221675x^{13} - 1611236x^{12} - 451827x^{11} + 2521879x^{10} + 278620x^{9} - 2250663x^{8} + 190394x^{7} + 981585x^{6} - 261440x^{5} - 142803x^{4} + 67700x^{3} - 6581x^{2} - 289x + 16$

Atkin–Lehner signs

$p$	$w_p$
691	$+1$

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
$\operatorname{tr} a_p$	−8	−5	−23	−6	−30	−12	−18	−15	−19	−83	−9	−18	−60	7	4	−72	−27	−19	6	−38	5	−10	−16	−32	24

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.229 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, 691]: [0, 58] (certified at ell = 2147480791)
T = [(2, 1), (3, 1)]; h of degree 57 = 1 x 57 (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(691, 2)[0]   # 691.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 691:2 1000