Atlas › Modular forms › Weight 2 › Level 671 › 671.2.a › 671.2.a.b

Newform orbit 671.2.a.b

Trace form

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

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

Coefficient field

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

$x^{6} + x^{5} - 12x^{4} + x^{3} + 19x^{2} + 7x - 1$

Atkin–Lehner signs

$p$	$w_p$
11	$+1$
61	$+1$

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
$\operatorname{tr} a_p$	0	−1	−1	−5	−6	−4	−5	−3	−3	−1	−10	−19	−7	−2	5	−9	−5	−6	−14	−14	−14	5	17	−25	−24

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.224 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, 11, 61, 671]: [0, 2, 5, 64] (certified at ell = 2147482921)
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(671, 2)[1]   # 671.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 671:2 1000