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

Newform orbit 817.2.a.f

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q + 6q^{2} + 2q^{3} + 22q^{4} + 7q^{5} - 2q^{6} + 13q^{7} + 15q^{8} + 24q^{9} + 15q^{11} - 11q^{12} + 4q^{13} - 3q^{14} + 19q^{15} + 18q^{16} + 25q^{17} + 5q^{18} + 18q^{19} + 10q^{20} - 6q^{21} + 7q^{22} + 44q^{23} - 30q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7} + 9T_{11}$ , whose minimal polynomial is

$x^{18} - 273x^{17} + 20638x^{16} + 738306x^{15} - 170972538x^{14} + 5339306270x^{13} + 261672202838x^{12} - 15966885491042x^{11} - 116890597414969x^{10} + 18541416974143211x^{9} - 22562199087086278x^{8} - 11942239270782113320x^{7} + 19525740860598216612x^{6} + 4557052065679112035435x^{5} + 8328233297492909126143x^{4} - 884408121105297968884897x^{3} - 5695788442236411468251141x^{2} + 37283500223327417027786714x + 313491175850637373695133786$

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
$\operatorname{tr} a_p$	6	2	7	13	15	4	25	18	44	2	4	−20	8	−18	51	14	14	−6	22	27	6	38	29	−12	−8

$\operatorname{tr} a_n$ 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)[5]   # 817.2.a.f

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

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