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

Newform orbit 671.2.a.d

Trace form

The 21 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$21q + 3q^{3} + 32q^{4} + 7q^{5} + 5q^{6} + 5q^{7} - 6q^{8} + 40q^{9} + q^{10} - 21q^{11} + 6q^{12} + 20q^{13} + 17q^{14} + 18q^{15} + 50q^{16} + q^{17} - 5q^{18} + 15q^{19} - 2q^{20} + 16q^{21} + 11q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 21. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{21} - 3x^{20} - 89x^{19} + 261x^{18} + 3104x^{17} - 8373x^{16} - 57470x^{15} + 131239x^{14} + 641963x^{13} - 1101240x^{12} - 4543308x^{11} + 4886972x^{10} + 20148283x^{9} - 9616526x^{8} - 52467170x^{7} + 480572x^{6} + 71279372x^{5} + 21215513x^{4} - 41473562x^{3} - 19610254x^{2} + 5027208x + 2168723$

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 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
$\operatorname{tr} a_p$	0	3	7	5	−21	20	1	15	11	−9	22	21	7	16	5	9	13	21	38	12	20	1	−19	37	68

$\operatorname{tr} a_n$ 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)[3]   # 671.2.a.d

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

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