Atlas › Modular forms › Weight 2 › Level 827 › 827.2.a › 827.2.a.d

Newform orbit 827.2.a.d

Trace form

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

$41q + 3q^{2} + 11q^{3} + 47q^{4} + 21q^{5} + 3q^{6} + 20q^{7} + 9q^{8} + 48q^{9} + 10q^{10} + 4q^{11} + 21q^{12} + 58q^{13} - 5q^{14} + 3q^{15} + 47q^{16} + 9q^{17} + 5q^{18} + 22q^{19} + 32q^{20} + 5q^{21} + q^{22} + 14q^{23} - 11q^{24} + O(q^{25})$

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

$x^{41} - 14x^{40} - 55x^{39} + 1585x^{38} - 1520x^{37} - 77111x^{36} + 223611x^{35} + 2099489x^{34} - 9078291x^{33} - 34637276x^{32} + 207233332x^{31} + 340895154x^{30} - 3065444496x^{29} - 1575636864x^{28} + 31130978322x^{27} - 5059693789x^{26} - 223247976930x^{25} + 133698946494x^{24} + 1145287003477x^{23} - 1021574016762x^{22} - 4213316571429x^{21} + 4604442594456x^{20} + 11030241166298x^{19} - 13610094704867x^{18} - 20127972946967x^{17} + 27062856301568x^{16} + 24541892129813x^{15} - 36031777739909x^{14} - 18294386337542x^{13} + 31272657946153x^{12} + 6463117653645x^{11} - 16704930200660x^{10} + 495753674719x^{9} + 4862675877976x^{8} - 1062563496283x^{7} - 564785340118x^{6} + 225856979606x^{5} + 182736257x^{4} - 9442907082x^{3} + 908414577x^{2} + 77806101x - 10387783$

Atkin–Lehner signs

$p$	$w_p$
827	$-1$

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
$\operatorname{tr} a_p$	3	11	21	20	4	58	9	22	14	6	37	92	11	19	19	6	−15	79	26	−16	99	0	4	−10	127

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.362 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, 827]: [0, 70] (certified at ell = 2147481883)
T = [(2, 1), (3, 1)]; h of degree 69 = 1 x 69 (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(827, 2)[3]   # 827.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 827:2 1000