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

Newform orbit 671.2.a.c

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q + 5q^{2} + 23q^{4} + q^{6} + 9q^{7} + 9q^{8} + 29q^{9} + 7q^{10} + 19q^{11} - 4q^{12} + 8q^{13} - 11q^{14} - 5q^{15} + 31q^{16} + 9q^{17} + 10q^{18} + 17q^{19} - 6q^{20} + 18q^{21} + 5q^{22} - 10q^{23} + 26q^{24} + O(q^{25})$

Coefficient field

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

$x^{19} - 5x^{18} - 62x^{17} + 332x^{16} + 1304x^{15} - 7837x^{14} - 10914x^{13} + 81680x^{12} + 35317x^{11} - 397276x^{10} - 68313x^{9} + 911444x^{8} + 306508x^{7} - 997474x^{6} - 652748x^{5} + 327694x^{4} + 432936x^{3} + 127619x^{2} + 1964x - 3011$

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
$\operatorname{tr} a_p$	5	0	0	9	19	8	9	17	−10	27	7	20	19	20	−19	3	−28	−19	3	−19	20	69	1	0	21

$\operatorname{tr} a_n$ 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)[2]   # 671.2.a.c

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

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