Atlas › Modular forms › Weight 2 › Level 817 › 817.2.a › 817.2.a.e

Newform orbit 817.2.a.e

Trace form

The 15 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 15. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ , whose minimal polynomial is

$x^{15} - 264x^{14} + 26675x^{13} - 1123606x^{12} - 1558291x^{11} + 1889225992x^{10} - 57157875449x^{9} - 184482654002x^{8} + 39346544381796x^{7} - 577474109462884x^{6} - 4628930440207026x^{5} + 185285795158774384x^{4} - 1187540459064453816x^{3} - 6154624214009754011x^{2} + 85315706702430857000x - 204676932582423476834$

Atkin–Lehner signs

$p$	$w_p$
19	$+1$
43	$-1$

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
$\operatorname{tr} a_p$	9	6	3	15	13	6	−3	−15	18	22	10	24	10	15	11	24	16	−10	4	53	−12	−20	11	38	−34

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.412 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, 19, 43, 817]: [0, 2, 4, 74] (certified at ell = 2147483647)
T = [(2, 1), (3, 3), (5, 5), (7, 7), (11, 9)]; h of degree 63 = 1 x 63 (phi(ord eps) x dim S_k^new); 15 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(817, 2)[4]   # 817.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 817:2 1000