Atlas › Modular forms › Weight 2 › Level 813 › 813.2.a › 813.2.a.f

Newform orbit 813.2.a.f

Trace form

The 17 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$17q - q^{2} + 17q^{3} + 21q^{4} + 4q^{5} - q^{6} + 16q^{7} - 3q^{8} + 17q^{9} + 5q^{11} + 21q^{12} + 17q^{13} - 3q^{14} + 4q^{15} + 33q^{16} - q^{18} + 31q^{19} + 3q^{20} + 16q^{21} - 8q^{22} + 6q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

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

$x^{17} - 3x^{16} - 84x^{15} + 292x^{14} + 2535x^{13} - 9977x^{12} - 35387x^{11} + 163348x^{10} + 226963x^{9} - 1404139x^{8} - 382662x^{7} + 6264450x^{6} - 2316327x^{5} - 12721183x^{4} + 9268740x^{3} + 7532359x^{2} - 5041565x - 1854497$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
271	$+1$

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	17	4	16	5	17	0	31	6	−4	24	14	−18	29	11	−1	−6	5	57	−5	40	−2	31	−18	42

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.246 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, 271, 813]: [0, 1, 23, 92] (certified at ell = 2147482801)
T = [(2, 1), (5, 1)]; h of degree 45 = 1 x 45 (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(813, 2)[5]   # 813.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 813:2 1000