Atlas › Modular forms › Weight 2 › Level 997 › 997.2.a › 997.2.a.h

Newform orbit 997.2.a.h

Trace form

The 42 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$42q + 14q^{2} + 24q^{3} + 48q^{4} + 16q^{5} + q^{6} + 15q^{7} + 30q^{8} + 56q^{9} - 7q^{10} + 49q^{11} + 32q^{12} + 9q^{13} + 11q^{14} + 17q^{15} + 48q^{16} + 37q^{17} + 14q^{18} + 15q^{19} + 3q^{20} - 6q^{21} + 14q^{22} + 68q^{23} + 4q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 42. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{7} + 7T_{11} + 9T_{13}$ , whose minimal polynomial is

$x^{42} - 585x^{41} + 132066x^{40} - 11373222x^{39} - 623054164x^{38} + 218135782298x^{37} - 14346610183473x^{36} - 616844733683896x^{35} + 137581580428822056x^{34} - 4925693267975406542x^{33} - 368885546398740662680x^{32} + 36500779723890717664722x^{31} - 328433786834578905236443x^{30} - 89885593433005491250968872x^{29} + 3846507899578002378626492055x^{28} + 67099086240867526894563443802x^{27} - 8655065485640679527021601460272x^{26} + 113717175831079936355090904725126x^{25} + 8739060562868840274851102120986612x^{24} - 303075277518135522322354569778467939x^{23} - 2916878179403808766903613232351272233x^{22} + 294278797201594027507558308589457083803x^{21} - 2102809208876772477047256728450248279729x^{20} - 148270611109600128775694888254463802417386x^{19} + 2610064537490180035607986565576031291517422x^{18} + 39661554676293041442913542527747989228457312x^{17} - 1226562918245586616271158599498427373036920887x^{16} - 4529413937282117713692385950686047375510308539x^{15} + 329392472331786667658225457202055956514874301069x^{14} - 244523777108577761353594065421902388772443152586x^{13} - 55764582900106702370156310805264898273155841808784x^{12} + 125793756686188989507775480725117236821960626606372x^{11} + 6069037882488091394449065152118252686824302537907828x^{10} - 11792058312741163751129593671178476090641042060061420x^{9} - 396643435855465459875895292506465330601768091543242340x^{8} + 262312522835133072849269110264657524249456187908519992x^{7} + 12265877041595088829962490282073858974429779959152386536x^{6} - 1713539306938268417776446761710411450294566372163654984x^{5} - 162760552500091277761574703495469247775813307876281877424x^{4} + 16936282187543586083420936497146729698404724865291226688x^{3} + 744739661178482844886653247794549557130581869840079416160x^{2} - 213567872376636993418337056046253662884077078905903130144x - 7078839275233764470708298558485493326232925434988564224$

Atkin–Lehner signs

$p$	$w_p$
997	$-1$

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
$\operatorname{tr} a_p$	14	24	16	15	49	9	37	15	68	31	18	28	18	51	49	65	53	−6	57	47	−19	30	46	9	8

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.719 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, 997]: [0, 83] (certified at ell = 2147475601)
T = [(2, 1), (3, 3), (7, 5), (11, 7), (13, 9)]; h of degree 82 = 1 x 82 (phi(ord eps) x dim S_k^new); 20 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(997, 2)[7]   # 997.2.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 997:2 1000