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

Newform orbit 749.2.a.b

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} + 5x^{5} + x^{4} - 16x^{3} + x^{2} + 11x - 4$

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
$\operatorname{tr} a_p$	−1	−4	0	6	−13	−5	1	−18	0	0	−15	−25	17	−38	5	−5	−14	6	−20	−26	−7	−22	−36	5	46

$\operatorname{tr} a_n$ 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)[1]   # 749.2.a.b

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

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