Atlas › Modular forms › Weight 2 › Level 667 › 667.2.a › 667.2.a.d

Newform orbit 667.2.a.d

Trace form

The 16 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 16. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{16} - 8x^{15} - 36x^{14} + 415x^{13} + 67x^{12} - 7173x^{11} + 7661x^{10} + 54271x^{9} - 85937x^{8} - 192086x^{7} + 352855x^{6} + 284971x^{5} - 550741x^{4} - 97488x^{3} + 162524x^{2} + 4832x - 11584$

Atkin–Lehner signs

$p$	$w_p$
23	$-1$
29	$+1$

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
$\operatorname{tr} a_p$	3	5	16	1	4	15	20	−4	16	−16	0	5	7	−17	29	63	11	0	−13	−23	−38	−27	36	−16	−30

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.215 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, 23, 29, 667]: [0, 3, 3, 62] (certified at ell = 2147482877)
T = [(2, 1), (3, 1)]; h of degree 51 = 1 x 51 (phi(ord eps) x dim S_k^new); 5 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(667, 2)[3]   # 667.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 667:2 1000