Atlas › Modular forms › Weight 2 › Level 670 › 670.2.a › 670.2.a.f

Newform orbit 670.2.a.f

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{2})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7}$ , whose minimal polynomial is

$x^{2} + 22x + 113$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$-1$
67	$+1$

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
$\operatorname{tr} a_p$	2	−4	2	−6	−10	−4	0	−4	0	0	8	−14	0	−16	−4	−16	0	10	−2	10	4	4	2	−14	−18

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.581 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, 5, 10, 67, 134, 335, 670]: [0, 1, 1, 3, 6, 19, 36, 106] (certified at ell = 2147483053)
T = [(3, 1), (7, 3)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 4 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(670, 2)[5]   # 670.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 670:2 1000