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

Newform orbit 745.2.a.e

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 4q^{2} + 10q^{3} + 14q^{4} + 14q^{5} - q^{6} + 11q^{7} + 6q^{8} + 18q^{9} + 4q^{10} + 11q^{11} + 10q^{12} + 5q^{13} + 3q^{14} + 10q^{15} + 10q^{16} + 5q^{17} + 5q^{18} - 5q^{19} + 14q^{20} - 5q^{21} - 2q^{22} + 31q^{23} - 22q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{14} - 34x^{13} + 290x^{12} + 2172x^{11} - 43039x^{10} + 69594x^{9} + 1778759x^{8} - 7308428x^{7} - 28357831x^{6} + 169048842x^{5} + 177581081x^{4} - 1509242984x^{3} - 582496869x^{2} + 4772186886x + 2481836260$

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 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	10	14	11	11	5	5	−5	31	17	−2	13	−7	10	22	27	4	−25	40	39	3	2	29	11	−20

$\operatorname{tr} a_n$ 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)[4]   # 745.2.a.e

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

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