Atlas › Modular forms › Weight 2 › Level 641 › 641.2.a › 641.2.a.b

Newform orbit 641.2.a.b

Trace form

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

$33q + 7q^{3} + 44q^{4} + q^{5} + 11q^{6} + 28q^{7} + 36q^{9} + 21q^{10} + 6q^{11} + 12q^{12} + 12q^{13} - 2q^{15} + 58q^{16} + 3q^{17} - 12q^{18} + 42q^{19} - 17q^{20} + 5q^{21} - 8q^{22} + 11q^{23} + 19q^{24} + O(q^{25})$

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

$x^{33} - 7x^{32} - 109x^{31} + 844x^{30} + 4793x^{29} - 43066x^{28} - 108176x^{27} + 1227201x^{26} + 1280303x^{25} - 21791939x^{24} - 5816382x^{23} + 255531689x^{22} - 39404004x^{21} - 2043980421x^{20} + 740387445x^{19} + 11326101189x^{18} - 4833523791x^{17} - 43503821109x^{16} + 16893852890x^{15} + 114097105530x^{14} - 31680541284x^{13} - 197470146947x^{12} + 24568695297x^{11} + 212731611627x^{10} + 9068601522x^{9} - 130249412361x^{8} - 27034348035x^{7} + 38028669128x^{6} + 14209959484x^{5} - 2757904350x^{4} - 1947828035x^{3} - 264562436x^{2} - 4314432x + 731243$

Atkin–Lehner signs

$p$	$w_p$
641	$-1$

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
$\operatorname{tr} a_p$	0	7	1	28	6	12	3	42	11	0	81	5	9	25	35	−22	11	57	13	11	18	108	7	−32	3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.22 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, 641]: [0, 54] (certified at ell = 2147479681)
T = [(2, 1), (3, 1)]; h of degree 53 = 1 x 53 (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(641, 2)[1]   # 641.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 641:2 1000