Atlas › Modular forms › Weight 2 › Level 773 › 773.2.a › 773.2.a.c

Newform orbit 773.2.a.c

Trace form

The 38 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$38q + 7q^{2} + 13q^{3} + 47q^{4} + 2q^{5} + 3q^{6} + 31q^{7} + 21q^{8} + 45q^{9} + 2q^{10} + 17q^{11} + 11q^{12} + 25q^{13} - 3q^{14} + 9q^{15} + 61q^{16} + 6q^{17} + 21q^{18} + 47q^{19} - 3q^{20} + 3q^{21} + 11q^{22} + 10q^{23} - 11q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 38. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ , whose minimal polynomial is

$x^{38} - 273x^{37} + 24527x^{36} - 43031x^{35} - 135418364x^{34} + 7458258841x^{33} + 118236088598x^{32} - 24276714698577x^{31} + 527237248687023x^{30} + 28917583198457157x^{29} - 1511338516714563694x^{28} - 3903993281186300096x^{27} + 1718964557708717010313x^{26} - 26038806269253029794262x^{25} - 977359999036290250191090x^{24} + 31541417401471284483774325x^{23} + 196693884488117656977592818x^{22} - 18629460569625279807947781937x^{21} + 96675450403103865095712540244x^{20} + 6464273739105764490231039516559x^{19} - 84839062358268252734080936249173x^{18} - 1303733267604666398569496919745814x^{17} + 30002834203807757862174738646535010x^{16} + 116869123111844004604138424282206266x^{15} - 6232689776538360823113904982636037531x^{14} + 8299489932770326052304313650198346943x^{13} + 813641997362244301164816511421842813079x^{12} - 3497125820998165559009338377853656846654x^{11} - 67576081274197722167024569826661506841215x^{10} + 406723104914326561591808010870941860693852x^{9} + 3592940299948764210378797434837229330920638x^{8} - 23748608996041177813987395767943813155230763x^{7} - 123832853145113120967552512381736790901326669x^{6} + 707655105478009750723877295571146942124056581x^{5} + 2615693596020483698695637366828978098132224251x^{4} - 9017686881782041376154274771176799286023793755x^{3} - 23183726753761948214324562943782648609155379961x^{2} + 41031466720428474967536967951873889624133433960x + 42658788046281151520920882464148305345922300236$

Atkin–Lehner signs

$p$	$w_p$
773	$-1$

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
$\operatorname{tr} a_p$	7	13	2	31	17	25	6	47	10	4	20	23	−7	119	12	0	15	1	134	−10	40	40	31	−11	9

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.297 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, 773]: [0, 65] (certified at ell = 2147480893)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 64 = 1 x 64 (phi(ord eps) x dim S_k^new); 14 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(773, 2)[2]   # 773.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 773:2 1000