Atlas › Modular forms › Weight 2 › Level 662 › 662.2.a › 662.2.a.d

Newform orbit 662.2.a.d

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5} + 5T_{7} + 7T_{11}$ , whose minimal polynomial is

$x^{9} - 16x^{8} - 4003x^{7} + 95158x^{6} + 3000690x^{5} - 71664412x^{4} - 587795154x^{3} + 14015379500x^{2} - 10320674300x - 71627755125$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
331	$-1$

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
$\operatorname{tr} a_p$	−9	6	5	6	−5	20	3	4	−1	11	8	16	−11	15	−3	13	0	15	39	−11	20	22	−4	−27	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 0.264 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, 2, 331, 662]: [0, 1, 28, 85] (certified at ell = 2147482921)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 7 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(662, 2)[3]   # 662.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 662:2 1000