Atlas › Modular forms › Weight 2 › Level 829 › 829.2.a › 829.2.a.c

Newform orbit 829.2.a.c

Trace form

The 39 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$39q + 9q^{2} + 7q^{3} + 45q^{4} + 7q^{5} + q^{6} + 6q^{7} + 24q^{8} + 54q^{9} - q^{10} + 59q^{11} - 3q^{13} + 22q^{14} + 4q^{15} + 49q^{16} + 15q^{17} + 5q^{18} + 26q^{19} + 6q^{20} + 5q^{21} + 6q^{22} + 34q^{23} - 2q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 39. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{39} - 16x^{38} - 20x^{37} + 1600x^{36} - 5170x^{35} - 61654x^{34} + 360419x^{33} + 1065653x^{32} - 10964792x^{31} - 3380249x^{30} + 191289760x^{29} - 199332954x^{28} - 2075548331x^{27} + 4148486911x^{26} + 14262056547x^{25} - 42185681477x^{24} - 59644020508x^{23} + 265098099625x^{22} + 122748765596x^{21} - 1097781663839x^{20} + 84807718110x^{19} + 3060150156757x^{18} - 1278937709651x^{17} - 5749345263883x^{16} + 3732662562311x^{15} + 7172365277621x^{14} - 5865355550830x^{13} - 5741257547544x^{12} + 5466969005195x^{11} + 2771606512484x^{10} - 2979227613278x^{9} - 729409974124x^{8} + 879443876850x^{7} + 93584701421x^{6} - 123021792480x^{5} - 6702528485x^{4} + 7082573422x^{3} + 156110780x^{2} - 125441967x + 3471453$

Atkin–Lehner signs

$p$	$w_p$
829	$-1$

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	7	7	6	59	−3	15	26	34	36	6	−10	30	35	31	2	70	−9	12	18	−23	37	83	11	−23

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.261 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, 829]: [0, 69] (certified at ell = 2147479273)
T = [(2, 1), (3, 1)]; h of degree 68 = 1 x 68 (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(829, 2)[2]   # 829.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 829:2 1000