Atlas › Modular forms › Weight 2 › Level 823 › 823.2.a › 823.2.a.a

Newform orbit 823.2.a.a

Trace form

The 30 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$30q - 10q^{2} - 12q^{3} + 28q^{4} - 20q^{5} - 6q^{6} - 14q^{7} - 27q^{8} + 18q^{9} - 11q^{10} - 16q^{11} - 36q^{12} - 17q^{13} - 15q^{14} - 5q^{15} + 28q^{16} - 77q^{17} - 21q^{18} - 6q^{19} - 31q^{20} - 14q^{21} - q^{22} - 27q^{23} + 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 30. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{30} + 22x^{29} + 150x^{28} - 81x^{27} - 5318x^{26} - 15963x^{25} + 57120x^{24} + 350064x^{23} - 80780x^{22} - 3378942x^{21} - 3107763x^{20} + 18188679x^{19} + 29356703x^{18} - 58559088x^{17} - 131202266x^{16} + 113488365x^{15} + 348749104x^{14} - 123603356x^{13} - 589424272x^{12} + 50714910x^{11} + 647512172x^{10} + 40629878x^{9} - 459693711x^{8} - 66269803x^{7} + 203074243x^{6} + 37411810x^{5} - 50752631x^{4} - 10008201x^{3} + 5552326x^{2} + 1051834x - 35531$

Atkin–Lehner signs

$p$	$w_p$
823	$+1$

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 odd 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
$\operatorname{tr} a_p$	−10	−12	−20	−14	−16	−17	−77	−6	−27	−37	−1	−6	−75	−15	−37	−44	−1	−1	10	9	−49	−3	−33	−78	−65

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.274 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, 823]: [0, 69] (certified at ell = 2147477467)
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(823, 2)[0]   # 823.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 823:2 1000