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

Newform orbit 667.2.a.b

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q - 3q^{2} - 3q^{3} + 11q^{4} - 16q^{5} - 2q^{6} - 7q^{7} - 9q^{8} + 3q^{9} - 6q^{11} - 21q^{12} - 15q^{13} - 8q^{14} + 6q^{15} + 17q^{16} - 18q^{17} - 12q^{18} - 6q^{19} - 39q^{20} - q^{21} - 5q^{22} - 12q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} + 6x^{11} - 17x^{10} - 109x^{9} + 134x^{8} + 668x^{7} - 689x^{6} - 1459x^{5} + 1850x^{4} + 509x^{3} - 1553x^{2} + 752x - 116$

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 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
$\operatorname{tr} a_p$	−3	−3	−16	−7	−6	−15	−18	−6	−12	−12	16	−1	3	−23	−35	−45	−11	4	−19	19	10	17	−12	−20	−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 0.215 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)[1]   # 667.2.a.b

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

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