Atlas › Modular forms › Weight 2 › Level 668 › 668.2.a › 668.2.a.c

Newform orbit 668.2.a.c

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 4q^{3} - 2q^{5} - 12q^{7} + 7q^{9} - 7q^{11} - 9q^{13} - 17q^{15} - q^{17} - 11q^{19} - 4q^{21} - 19q^{23} + O(q^{25})$

Coefficient field

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

$x^{7} + 6x^{6} + 2x^{5} - 44x^{4} - 94x^{3} - 69x^{2} - 18x - 1$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 668$ )
167	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

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
$\operatorname{tr} a_p$	0	−4	−2	−12	−7	−9	−1	−11	−19	−5	−13	−26	−2	−24	−11	4	−4	−5	−42	9	27	−8	16	9	−8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.214 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, 4, 167, 334, 668]: [0, 1, 2, 15, 44, 87] (certified at ell = 2147483237)
T = [(3, 1), (5, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 2 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(668, 2)[2]   # 668.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 668:2 1000