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

Newform orbit 745.2.a.b

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - 2q^{2} + 6q^{4} - 10q^{5} - 9q^{6} - q^{7} - 6q^{8} - 2q^{9} + 2q^{10} - 17q^{11} + 5q^{13} - 13q^{14} - 2q^{16} - 9q^{17} - q^{18} - 13q^{19} - 6q^{20} - 9q^{21} - 2q^{22} - 33q^{23} - 14q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{10} + 2x^{9} - 110x^{8} - 74x^{7} + 3916x^{6} + 1170x^{5} - 55901x^{4} - 29482x^{3} + 306371x^{2} + 281090x - 205524$

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
$\operatorname{tr} a_p$	−2	0	−10	−1	−17	5	−9	−13	−33	−23	−18	1	−19	14	−16	−3	−36	−13	20	−53	5	−6	21	−1	14

$\operatorname{tr} a_n$ 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)[1]   # 745.2.a.b

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

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