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

Newform orbit 691.2.a.b

Trace form

The 33 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$33q + 8q^{2} + q^{3} + 36q^{4} + 23q^{5} + 5q^{6} + 4q^{7} + 18q^{8} + 46q^{9} + 28q^{11} - 4q^{12} + 12q^{13} + 13q^{14} + 5q^{15} + 38q^{16} + 16q^{17} + 9q^{18} - q^{19} + 30q^{20} + 41q^{21} + 5q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 33. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{33} - 9x^{32} - 88x^{31} + 989x^{30} + 2658x^{29} - 45639x^{28} - 18458x^{27} + 1173792x^{26} - 809385x^{25} - 18815481x^{24} + 24706887x^{23} + 199174888x^{22} - 339009864x^{21} - 1438662908x^{20} + 2809079276x^{19} + 7213122021x^{18} - 15206721713x^{17} - 25189818947x^{16} + 55258293652x^{15} + 60441689914x^{14} - 135279780014x^{13} - 95614322796x^{12} + 220452398498x^{11} + 90485940325x^{10} - 232561988863x^{9} - 38471129804x^{8} + 149890652651x^{7} - 5873629243x^{6} - 52035311350x^{5} + 10426531406x^{4} + 7043802576x^{3} - 2185982336x^{2} - 28139248x + 18569312$

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 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
$\operatorname{tr} a_p$	8	1	23	4	28	12	16	−1	5	91	−1	24	46	−3	10	56	3	27	−8	60	−9	16	10	46	−2

$\operatorname{tr} a_n$ 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)[1]   # 691.2.a.b

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

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