Atlas › Modular forms › Weight 2 › Level 383 › 383.2.a › 383.2.a.c

Newform orbit 383.2.a.c

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q + 5q^{2} + 2q^{3} + 29q^{4} + 3q^{5} + q^{6} + 17q^{7} + 15q^{8} + 34q^{9} + q^{10} - 7q^{12} + 28q^{13} - 8q^{14} - 2q^{15} + 35q^{16} + 16q^{17} + 19q^{18} + 13q^{19} - 4q^{20} - 4q^{21} + 12q^{22} + 7q^{23} - 11q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{24} - 7x^{23} - 68x^{22} + 547x^{21} + 1699x^{20} - 17354x^{19} - 18019x^{18} + 295509x^{17} + 27978x^{16} - 2989084x^{15} + 1171334x^{14} + 18749018x^{13} - 11850892x^{12} - 74160017x^{11} + 52592868x^{10} + 185125453x^{9} - 123483693x^{8} - 283928163x^{7} + 152509261x^{6} + 247140619x^{5} - 86979981x^{4} - 99950623x^{3} + 15551343x^{2} + 9163522x - 1592767$

Atkin–Lehner signs

$p$	$w_p$
383	$-1$

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
$\operatorname{tr} a_p$	5	2	3	17	0	28	16	13	7	−2	0	35	0	34	−3	−9	−6	3	42	−13	83	−3	−23	−38	63

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.183 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, 383]: [0, 33] (certified at ell = 2147476031)
T = [(2, 1), (3, 1)]; h of degree 32 = 1 x 32 (phi(ord eps) x dim S_k^new); 4 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(383, 2)[2]   # 383.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 383:2 1000