Atlas › Modular forms › Weight 2 › Level 743 › 743.2.a › 743.2.a.b

Newform orbit 743.2.a.b

Trace form

The 41 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$41q + 3q^{2} + 11q^{3} + 47q^{4} + 12q^{5} + 5q^{6} + 22q^{7} + 9q^{8} + 58q^{9} + 13q^{10} + 5q^{11} + 26q^{12} + 34q^{13} - 6q^{14} - q^{15} + 47q^{16} + 26q^{17} + 6q^{18} + 12q^{19} + 16q^{20} - q^{21} - 8q^{22} + 10q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 41. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{41} - 14x^{40} - 62x^{39} + 1680x^{38} - 1174x^{37} - 86844x^{36} + 230944x^{35} + 2545175x^{34} - 10178588x^{33} - 46493528x^{32} + 250346361x^{31} + 539909484x^{30} - 4021791553x^{29} - 3707722816x^{28} + 44930048043x^{27} + 8579288415x^{26} - 359769189093x^{25} + 98924943648x^{24} + 2094162757011x^{23} - 1214086681000x^{22} - 8898514245448x^{21} + 6877993014323x^{20} + 27530982862938x^{19} - 24149143133837x^{18} - 61579147928303x^{17} + 55365553207990x^{16} + 98610032779687x^{15} - 82444137487698x^{14} - 111661269368849x^{13} + 76568827025919x^{12} + 87476628213712x^{11} - 40431278158507x^{10} - 45088099660554x^{9} + 9352318132162x^{8} + 13642610937882x^{7} + 340754120220x^{6} - 1851961089579x^{5} - 369249587015x^{4} + 34284952976x^{3} + 11987932992x^{2} - 58099941x - 105718473$

Atkin–Lehner signs

$p$	$w_p$
743	$-1$

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
$\operatorname{tr} a_p$	3	11	12	22	5	34	26	12	10	−8	37	62	30	20	31	10	−10	31	18	−1	171	1	0	10	190

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.248 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, 743]: [0, 63] (certified at ell = 2147477851)
T = [(2, 1), (3, 1)]; h of degree 62 = 1 x 62 (phi(ord eps) x dim S_k^new); 6 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(743, 2)[1]   # 743.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 743:2 1000