Atlas › Modular forms › Weight 2 › Level 721 › 721.2.a › 721.2.a.e

Newform orbit 721.2.a.e

Trace form

The 12 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$12q + 2q^{2} - 3q^{3} + 12q^{4} + q^{5} + 8q^{6} - 12q^{7} + 6q^{8} + 23q^{9} + 14q^{10} + 18q^{11} - 16q^{12} - 9q^{13} - 2q^{14} + 17q^{15} + 8q^{16} - 9q^{17} + 19q^{18} - 14q^{19} + 12q^{20} + 3q^{21} - 2q^{22} + 30q^{23} + 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 12. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{12} + x^{11} - 55x^{10} - 51x^{9} + 955x^{8} + 1135x^{7} - 6478x^{6} - 9936x^{5} + 14180x^{4} + 27011x^{3} - 4536x^{2} - 18904x - 4855$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
103	$-1$

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
$\operatorname{tr} a_p$	2	−3	1	−12	18	−9	−9	−14	30	−6	31	−2	7	23	34	3	11	5	20	76	−14	14	−10	−10	−15

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.226 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, 7, 103, 721]: [0, 1, 9, 70] (certified at ell = 2147482501)
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(721, 2)[4]   # 721.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 721:2 1000