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

Newform orbit 883.2.a.b

Trace form

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

$39q + 10q^{2} + 7q^{3} + 40q^{4} + 29q^{5} + q^{6} + 6q^{7} + 27q^{8} + 42q^{9} + 10q^{10} + 5q^{11} + 5q^{12} + 35q^{13} + 13q^{14} + 2q^{15} + 34q^{16} + 55q^{17} + 16q^{18} + 6q^{19} + 48q^{20} + 19q^{21} + 22q^{23} - 6q^{24} + O(q^{25})$

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

$x^{39} - 17x^{38} + 5x^{37} + 1449x^{36} - 6259x^{35} - 46505x^{34} + 348116x^{33} + 567429x^{32} - 9248558x^{31} + 4061831x^{30} + 143884438x^{29} - 232571441x^{28} - 1400298875x^{27} + 3573644372x^{26} + 8659597980x^{25} - 31231778871x^{24} - 33143698872x^{23} + 176890239533x^{22} + 69248388465x^{21} - 685278976267x^{20} - 24584330025x^{19} + 1869682604207x^{18} - 269892681284x^{17} - 3646425019621x^{16} + 782760569651x^{15} + 5094324732701x^{14} - 1096159092829x^{13} - 5043505680096x^{12} + 892099260886x^{11} + 3458098683647x^{10} - 431332365586x^{9} - 1587993501599x^{8} + 119547367736x^{7} + 465920630566x^{6} - 17338920611x^{5} - 81318309328x^{4} + 1122282473x^{3} + 7452433768x^{2} - 24350032x - 264894784$

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 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
$\operatorname{tr} a_p$	10	7	29	6	5	35	55	6	22	52	−2	20	30	−8	31	96	18	36	−15	9	16	−16	65	38	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)[1]   # 883.2.a.b

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

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