Atlas › Modular forms › Weight 2 › Level 663 › 663.2.a › 663.2.a.i

Newform orbit 663.2.a.i

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11} + 9T_{19}$ , whose minimal polynomial is

$x^{6} - 46x^{5} - 3533x^{4} - 2592x^{3} + 997023x^{2} - 1010264x - 43523461$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
13	$-1$
17	$-1$

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	6	4	2	8	6	6	−4	0	−4	−10	2	12	−16	14	8	18	−12	−8	12	6	−8	6	26	−2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.425 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, 3, 13, 17, 39, 51, 221, 663]: [0, 1, 1, 2, 6, 8, 22, 88] (certified at ell = 2147483137)
T = [(2, 1), (5, 3), (7, 5), (11, 7), (19, 9)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 8 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(663, 2)[8]   # 663.2.a.i

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

cargo run --release -p sagebrush-web --example atlas -- 663:2 1000