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

Newform orbit 649.2.a.c

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{4} + 71x^{3} + 97x^{2} - 52561x - 450239$

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 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
$\operatorname{tr} a_p$	1	1	−5	−4	4	−4	−3	−6	−23	−4	−11	2	9	5	22	−6	4	−7	−4	−38	29	−7	−8	−5	7

$\operatorname{tr} a_n$ 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)[2]   # 649.2.a.c

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

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