Atlas › Modular forms › Weight 2 › Level 983 › 983.2.a › 983.2.a.b

Newform orbit 983.2.a.b

Trace form

The 54 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$54q + 8q^{2} + 6q^{3} + 64q^{4} + 7q^{5} + 5q^{6} + 31q^{7} + 24q^{8} + 74q^{9} + 15q^{10} + 8q^{11} + 10q^{12} + 34q^{13} + q^{14} + 3q^{15} + 80q^{16} + 32q^{17} + 28q^{18} + 15q^{19} + 4q^{20} + 11q^{21} + 25q^{22} + 15q^{23} - 7q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 54. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{54} - 14x^{53} - 111x^{52} + 2360x^{51} + 2697x^{50} - 179089x^{49} + 225589x^{48} + 8104785x^{47} - 21281448x^{46} - 244049827x^{45} + 891850160x^{44} + 5159234525x^{43} - 23864025223x^{42} - 78581188962x^{41} + 451379264271x^{40} + 867570562965x^{39} - 6324281642990x^{38} - 6825910521764x^{37} + 67370787677215x^{36} + 35850255940786x^{35} - 554416492643671x^{34} - 95295601131327x^{33} + 3560525262951157x^{32} - 206628357147179x^{31} - 17959537702513957x^{30} + 3405747145593989x^{29} + 71400021337438578x^{28} - 17335004273542240x^{27} - 223840006225043853x^{26} + 51876393824062711x^{25} + 551531427544872129x^{24} - 93140137214254004x^{23} - 1058914010115507790x^{22} + 70075738080758454x^{21} + 1558905418127889830x^{20} + 101159484363082524x^{19} - 1712449067749539690x^{18} - 369332153130280601x^{17} + 1340596865295621682x^{16} + 516418198587860271x^{15} - 686742556317901656x^{14} - 408525933160076552x^{13} + 186578051903695975x^{12} + 182816976227886889x^{11} - 3668011888340525x^{10} - 39074157989857628x^{9} - 9474784599070369x^{8} + 1869176801343401x^{7} + 989437488258931x^{6} + 61042445704814x^{5} - 19902464268304x^{4} - 2349593492964x^{3} + 71136808273x^{2} + 17371375388x + 533000719$

Atkin–Lehner signs

$p$	$w_p$
983	$-1$

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
$\operatorname{tr} a_p$	8	6	7	31	8	34	32	15	15	8	16	80	30	53	22	10	−4	24	54	−6	172	28	14	−5	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 0.308 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, 983]: [0, 83] (certified at ell = 2147475899)
T = [(2, 1), (3, 1)]; h of degree 82 = 1 x 82 (phi(ord eps) x dim S_k^new); 8 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(983, 2)[1]   # 983.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 983:2 1000