Atlas › Modular forms › Weight 2 › Level 745 › 745.2.a › 745.2.a.c

Newform orbit 745.2.a.c

Trace form

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

$11q - 5q^{2} - 12q^{3} + 9q^{4} + 11q^{5} + 3q^{6} - 13q^{7} - 15q^{8} + 11q^{9} - 5q^{10} - 9q^{11} - 20q^{12} - 15q^{13} - 7q^{14} - 12q^{15} + 13q^{16} - 9q^{17} + 2q^{18} - q^{19} + 9q^{20} + 11q^{21} - 10q^{22} - 43q^{23} + 14q^{24} + O(q^{25})$

Coefficient field

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

$x^{11} + 41x^{10} + 618x^{9} + 3570x^{8} - 4180x^{7} - 135206x^{6} - 449891x^{5} + 194207x^{4} + 2938649x^{3} + 4336765x^{2} + 1813658x - 123316$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
149	$-1$

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
$\operatorname{tr} a_p$	−5	−12	11	−13	−9	−15	−9	−1	−43	−1	−2	−5	1	−26	−26	−35	−12	5	−24	−57	−7	−32	−25	7	−12

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.333 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, 5, 149, 745]: [0, 1, 13, 76] (certified at ell = 2147481629)
T = [(2, 1), (3, 3)]; h of degree 49 = 1 x 49 (phi(ord eps) x dim S_k^new); 7 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(745, 2)[2]   # 745.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 745:2 1000