Atlas › Modular forms › Weight 2 › Level 749 › 749.2.a › 749.2.a.c

Newform orbit 749.2.a.c

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q - 5q^{3} + 8q^{4} - 4q^{5} - 6q^{6} - 12q^{7} - 3q^{8} + 9q^{9} - 17q^{10} - 16q^{11} - 17q^{12} - 10q^{13} - 8q^{15} - 4q^{16} - 5q^{17} + 3q^{18} - 41q^{19} - 6q^{20} + 5q^{21} + q^{22} - 5q^{23} - 20q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} + 5x^{11} - 20x^{10} - 100x^{9} + 125x^{8} + 608x^{7} - 215x^{6} - 1466x^{5} - 105x^{4} + 1466x^{3} + 440x^{2} - 512x - 226$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
107	$+1$

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
$\operatorname{tr} a_p$	0	−5	−4	−12	−16	−10	−5	−41	−5	1	−25	2	−14	−20	2	6	−24	−29	−12	26	−17	−23	3	−18	−10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.233 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, 7, 107, 749]: [0, 1, 10, 74] (certified at ell = 2147482621)
T = [(2, 1), (3, 1)]; h of degree 53 = 1 x 53 (phi(ord eps) x dim S_k^new); 6 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(749, 2)[2]   # 749.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 749:2 1000