Atlas › Modular forms › Weight 2 › Level 967 › 967.2.a › 967.2.a.b

Newform orbit 967.2.a.b

Trace form

The 45 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$45q + 11q^{2} + 11q^{3} + 47q^{4} + 27q^{5} + 4q^{7} + 30q^{8} + 62q^{9} + 5q^{10} + 11q^{11} + 22q^{12} + 16q^{13} + 5q^{14} + 14q^{15} + 43q^{16} + 72q^{17} + 16q^{18} + 8q^{19} + 52q^{20} + 6q^{21} + 14q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 45. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{45} - 22x^{44} + 75x^{43} + 1750x^{42} - 14781x^{41} - 36713x^{40} + 777960x^{39} - 909371x^{38} - 20729174x^{37} + 66512095x^{36} + 308026081x^{35} - 1688941388x^{34} - 2202719072x^{33} + 24963132633x^{32} - 4430649564x^{31} - 238612202786x^{30} + 261470953889x^{29} + 1522140319437x^{28} - 2787045246124x^{27} - 6468617866370x^{26} + 16893390290002x^{25} + 17704002412342x^{24} - 66697994368742x^{23} - 28040636861569x^{22} + 179013239396914x^{21} + 14464858690701x^{20} - 331133562900364x^{19} + 32024124900740x^{18} + 420836492835677x^{17} - 71215891538934x^{16} - 360449807769639x^{15} + 61485701110546x^{14} + 200227218191068x^{13} - 25520835560203x^{12} - 67683897147843x^{11} + 4435064638067x^{10} + 12503892827110x^{9} - 223907061784x^{8} - 1038272531821x^{7} + 38637285331x^{6} + 35133870202x^{5} - 3204299072x^{4} - 195873060x^{3} + 28999222x^{2} - 677913x - 3717$

Atkin–Lehner signs

$p$	$w_p$
967	$-1$

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
$\operatorname{tr} a_p$	11	11	27	4	11	16	72	8	14	14	12	24	74	4	62	66	34	−1	−1	28	42	−30	17	146	83

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.305 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, 967]: [0, 81] (certified at ell = 2147480791)
T = [(2, 1), (3, 1)]; h of degree 80 = 1 x 80 (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(967, 2)[1]   # 967.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 967:2 1000