Atlas › Modular forms › Weight 2 › Level 681 › 681.2.a › 681.2.a.h

Newform orbit 681.2.a.h

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q - q^{2} - 10q^{3} + 15q^{4} - 4q^{5} + q^{6} + 12q^{7} + 10q^{9} - 3q^{10} - 3q^{11} - 15q^{12} + 10q^{13} + 6q^{14} + 4q^{15} + 29q^{16} - 8q^{17} - q^{18} + 25q^{19} + 7q^{20} - 12q^{21} - 10q^{22} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{10} + 30x^{9} - 11937x^{8} - 718059x^{7} + 5470496x^{6} + 909155003x^{5} + 2946518012x^{4} - 410210615465x^{3} - 1263728691306x^{2} + 66792247025182x - 97212942723497$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
227	$-1$

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
$\operatorname{tr} a_p$	−1	−10	−4	12	−3	10	−8	25	0	−3	37	13	7	16	26	−18	8	20	28	29	16	4	5	14	14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.404 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, 3, 227, 681]: [0, 1, 20, 78] (certified at ell = 2147479573)
T = [(2, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 10 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(681, 2)[7]   # 681.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 681:2 1000