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

Newform orbit 649.2.a.d

Trace form

The 11 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$11q + 4q^{2} + 2q^{3} + 8q^{4} - q^{5} + 9q^{6} + 4q^{7} + 12q^{8} + 11q^{9} + 2q^{10} - 11q^{11} + 12q^{12} + q^{13} + 14q^{14} + 5q^{15} - 2q^{16} + 5q^{17} + 20q^{18} + 6q^{19} + 7q^{20} - 4q^{21} - 4q^{22} + 26q^{23} - 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 11. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{11} - 82x^{10} - 12022x^{9} + 1184342x^{8} + 22800726x^{7} - 4042940087x^{6} + 32353834854x^{5} + 3885137217180x^{4} - 56028719873876x^{3} - 994061940277488x^{2} + 12983016841403232x + 67862976861413264$

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
$\operatorname{tr} a_p$	4	2	−1	4	−11	1	5	6	26	23	4	−26	18	17	35	4	11	0	−1	30	−8	10	22	−17	−19

$\operatorname{tr} a_n$ 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)[3]   # 649.2.a.d

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

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