Atlas › Modular forms › Weight 2 › Level 815 › 815.2.a › 815.2.a.f

Newform orbit 815.2.a.f

Trace form

The 20 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$20q + 5q^{2} + 6q^{3} + 27q^{4} - 20q^{5} + q^{6} + 11q^{7} + 18q^{8} + 32q^{9} - 5q^{10} - 5q^{11} + 14q^{12} + 23q^{13} - 10q^{14} - 6q^{15} + 45q^{16} + 21q^{17} + 32q^{18} + 11q^{19} - 27q^{20} - 4q^{21} + 15q^{22} + 12q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 20. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{20} - 11x^{19} - 20x^{18} + 557x^{17} - 575x^{16} - 10671x^{15} + 19592x^{14} + 101916x^{13} - 201949x^{12} - 559503x^{11} + 942588x^{10} + 1846111x^{9} - 2072493x^{8} - 3363255x^{7} + 2162424x^{6} + 3066091x^{5} - 1066497x^{4} - 1263704x^{3} + 215190x^{2} + 179029x - 5066$

Atkin–Lehner signs

$p$	$w_p$
5	$+1$
163	$-1$

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	6	−20	11	−5	23	21	11	12	−9	10	43	−19	36	9	19	−23	−5	23	−15	60	40	9	−26	44

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.339 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, 5, 163, 815]: [0, 1, 14, 84] (certified at ell = 2147482693)
T = [(2, 1), (3, 1)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 6 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(815, 2)[5]   # 815.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 815:2 1000