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

Newform orbit 667.2.a.c

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$13q + 4q^{2} + 3q^{3} + 12q^{4} + 16q^{5} + q^{7} + 6q^{8} + 14q^{9} + 10q^{10} + 10q^{11} + 3q^{12} + 7q^{13} - 12q^{14} + 8q^{15} + 2q^{16} + 26q^{17} + 7q^{18} + 25q^{20} + 11q^{21} - 15q^{22} - 13q^{23} + 10q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 13. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{13} - 7x^{12} - 21x^{11} + 216x^{10} + 49x^{9} - 2272x^{8} + 885x^{7} + 10546x^{6} - 4733x^{5} - 21991x^{4} + 7354x^{3} + 18207x^{2} - 2672x - 3616$

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
$\operatorname{tr} a_p$	4	3	16	1	10	7	26	0	−13	13	−6	15	9	1	15	43	−9	20	1	17	26	5	4	48	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)[2]   # 667.2.a.c

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

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