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

Newform orbit 749.2.a.e

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q + 2q^{2} + 3q^{3} + 32q^{4} - 2q^{5} + 20q^{7} + 3q^{8} + 29q^{9} + 3q^{10} + 16q^{11} - 5q^{12} + 2q^{13} + 2q^{14} + 52q^{16} - 3q^{17} - 3q^{18} + 41q^{19} - 22q^{20} + 3q^{21} + 9q^{22} + 11q^{23} + 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{20} - 5x^{19} - 68x^{18} + 370x^{17} + 1495x^{16} - 9130x^{15} - 14047x^{14} + 100304x^{13} + 70742x^{12} - 559175x^{11} - 232136x^{10} + 1601906x^{9} + 476205x^{8} - 2190936x^{7} - 296383x^{6} + 1392038x^{5} - 74673x^{4} - 325270x^{3} + 61456x^{2} + 9592x - 1186$

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 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
$\operatorname{tr} a_p$	2	3	−2	20	16	2	−3	41	11	9	7	38	4	40	12	14	2	15	44	10	−23	21	29	−14	−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 0.233 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)[4]   # 749.2.a.e

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

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