Atlas › Modular forms › Weight 2 › Level 649 › 649.2.a › 649.2.a.f

Newform orbit 649.2.a.f

Trace form

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

$17q + 3q^{2} + 4q^{3} + 23q^{4} + 7q^{5} - 5q^{6} + 4q^{7} + 9q^{8} + 29q^{9} - 4q^{10} + 17q^{11} + 2q^{12} - 11q^{13} + 16q^{14} + 20q^{15} + 39q^{16} + 6q^{17} - 15q^{18} - 4q^{19} - q^{20} - 11q^{21} + 3q^{22} + 36q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 17. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{17} - 12x^{16} - 26746x^{15} + 798505x^{14} + 260182258x^{13} - 10906681000x^{12} - 1173380969327x^{11} + 61598455616964x^{10} + 2517473168057563x^{9} - 167656485621870880x^{8} - 2217138230295877859x^{7} + 226986158331107788833x^{6} + 40809040809689999828x^{5} - 143073953131882365724796x^{4} + 746848332653814837936820x^{3} + 34771512514108240174900720x^{2} - 132654962130733971841453408x - 2999432250775016512729227120$

Atkin–Lehner signs

$p$	$w_p$
11	$-1$
59	$+1$

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
$\operatorname{tr} a_p$	3	4	7	4	17	−11	6	−4	36	13	6	14	6	−11	25	0	−17	−38	1	89	−16	34	−12	−5	−15

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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, 59, 649]: [0, 2, 6, 62] (certified at ell = 2147482481)
T = [(2, 1), (3, 3), (7, 5), (13, 7), (17, 9)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 12 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(649, 2)[5]   # 649.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 649:2 1000