Atlas › Modular forms › Weight 2 › Level 971 › 971.2.a › 971.2.a.b

Newform orbit 971.2.a.b

Trace form

The 55 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$55q + 4q^{2} + 4q^{3} + 76q^{4} + 8q^{5} + 9q^{6} + 14q^{7} + 9q^{8} + 79q^{9} + 12q^{10} + 12q^{11} + 8q^{12} + 44q^{13} + q^{14} + 114q^{16} + 11q^{17} + 8q^{18} + 28q^{19} - 10q^{20} + 53q^{21} + 11q^{22} + 7q^{23} + 13q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 55. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{55} - 8x^{54} - 192x^{53} + 1653x^{52} + 16467x^{51} - 155552x^{50} - 832490x^{49} + 8859951x^{48} + 27513533x^{47} - 342552565x^{46} - 620255808x^{45} + 9560336810x^{44} + 9542568603x^{43} - 200039004039x^{42} - 93479542287x^{41} + 3218125643891x^{40} + 381456721201x^{39} - 40508476201694x^{38} + 4188178170046x^{37} + 403903807198794x^{36} - 93725186018064x^{35} - 3216694666905702x^{34} + 932832887759030x^{33} + 20563759089771175x^{32} - 6049516113460207x^{31} - 105729683545596067x^{30} + 27359377150002232x^{29} + 436802720088623628x^{28} - 85803059029424205x^{27} - 1444722223339655247x^{26} + 171553326681309108x^{25} + 3802312480220463621x^{24} - 136637826187435892x^{23} - 7898183377210371285x^{22} - 343242990581223620x^{21} + 12821856658593940981x^{20} + 1447058216365319705x^{19} - 16086245699443563870x^{18} - 2696063266367872796x^{17} + 15401865488221484443x^{16} + 3151020663404426835x^{15} - 11091187643226348311x^{14} - 2459346459369150402x^{13} + 5899575793486099503x^{12} + 1282287035949473613x^{11} - 2262199897609493302x^{10} - 428839964626186720x^{9} + 603464765180898600x^{8} + 82225463481453372x^{7} - 105741192233558918x^{6} - 5918747055723043x^{5} + 10908247964279863x^{4} - 532820999319964x^{3} - 489878290830202x^{2} + 83074878499544x - 3818667782183$

Atkin–Lehner signs

$p$	$w_p$
971	$-1$

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
$\operatorname{tr} a_p$	4	4	8	14	12	44	11	28	7	62	16	56	7	46	−16	37	−18	146	15	−9	49	72	−12	17	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.315 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, 971]: [0, 82] (certified at ell = 2147479121)
T = [(2, 1), (3, 1)]; h of degree 81 = 1 x 81 (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(971, 2)[1]   # 971.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 971:2 1000