Atlas › Modular forms › Weight 2 › Level 809 › 809.2.a › 809.2.a.c

Newform orbit 809.2.a.c

Trace form

The 41 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$41q + 12q^{3} + 54q^{4} + 5q^{5} + 4q^{6} + 25q^{7} - 3q^{8} + 47q^{9} + 21q^{10} + 4q^{11} + 23q^{12} + 9q^{13} + 12q^{14} + 22q^{15} + 76q^{16} + 2q^{17} - 8q^{18} + 46q^{19} + 6q^{20} + 8q^{21} + 6q^{22} + 9q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 41. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{41} - 12x^{40} - 85x^{39} + 1522x^{38} + 1463x^{37} - 83488x^{36} + 97528x^{35} + 2616974x^{34} - 6174083x^{33} - 52139890x^{32} + 169843039x^{31} + 693751999x^{30} - 2871307182x^{29} - 6275828962x^{28} + 32938560235x^{27} + 38052291270x^{26} - 268186838836x^{25} - 142049469551x^{24} + 1585513941046x^{23} + 197111779751x^{22} - 6873268600216x^{21} + 1023215881055x^{20} + 21846306091797x^{19} - 7365776064193x^{18} - 50436387288817x^{17} + 23594719563267x^{16} + 82947339538547x^{15} - 46169484277041x^{14} - 94166535460630x^{13} + 57545578755006x^{12} + 70538309207495x^{11} - 44460267213982x^{10} - 33001374047549x^{9} + 19842525541917x^{8} + 9251314359131x^{7} - 4621576563705x^{6} - 1508903776063x^{5} + 473146481727x^{4} + 125872485363x^{3} - 14515439147x^{2} - 2643641296x + 210257611$

Atkin–Lehner signs

$p$	$w_p$
809	$-1$

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
$\operatorname{tr} a_p$	0	12	5	25	4	9	2	46	9	0	109	18	7	17	18	−4	19	57	17	54	12	128	10	5	38

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.263 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, 809]: [0, 68] (certified at ell = 2147474929)
T = [(2, 1), (3, 1)]; h of degree 67 = 1 x 67 (phi(ord eps) x dim S_k^new); 7 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(809, 2)[2]   # 809.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 809:2 1000