Atlas › Modular forms › Weight 2 › Level 883 › 883.2.a › 883.2.a.a

Newform orbit 883.2.a.a

Trace form

The 34 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$34q - 12q^{2} - 11q^{3} + 32q^{4} - 31q^{5} - 7q^{6} - 10q^{7} - 33q^{8} + 25q^{9} - 6q^{10} - 9q^{11} - 33q^{12} - 39q^{13} - 11q^{14} - 6q^{15} + 32q^{16} - 55q^{17} - 20q^{18} - 12q^{19} - 42q^{20} - 29q^{21} - 8q^{22} - 26q^{23} - 6q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 34. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{34} + 23x^{33} + 158x^{32} - 248x^{31} - 7691x^{30} - 21409x^{29} + 116345x^{28} + 671228x^{27} - 435559x^{26} - 8884927x^{25} - 7409337x^{24} + 66662800x^{23} + 115099584x^{22} - 306807106x^{21} - 789684792x^{20} + 859124481x^{19} + 3295805460x^{18} - 1245942160x^{17} - 9045096895x^{16} - 124423913x^{15} + 16650336808x^{14} + 4141447490x^{13} - 20356296268x^{12} - 7921397451x^{11} + 15957215007x^{10} + 7335554970x^{9} - 7514022698x^{8} - 3490291916x^{7} + 1907884166x^{6} + 752218553x^{5} - 216178720x^{4} - 57515162x^{3} + 4778720x^{2} + 1510472x + 72575$

Atkin–Lehner signs

$p$	$w_p$
883	$+1$

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
$\operatorname{tr} a_p$	−12	−11	−31	−10	−9	−39	−55	−12	−26	−54	0	−20	−36	−30	−29	−100	−32	−16	−17	3	−10	14	−53	−52	−5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.292 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, 883]: [0, 74] (certified at ell = 2147479489)
T = [(2, 1), (3, 1)]; h of degree 73 = 1 x 73 (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(883, 2)[0]   # 883.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 883:2 1000