Atlas › Modular forms › Weight 2 › Level 727 › 727.2.a › 727.2.a.b

Newform orbit 727.2.a.b

Trace form

The 36 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$36q + 14q^{2} + 6q^{3} + 40q^{4} + 6q^{5} + 3q^{6} + q^{7} + 39q^{8} + 54q^{9} - 4q^{10} + 9q^{11} + 10q^{12} + 16q^{13} + 12q^{14} + 4q^{15} + 44q^{16} + 63q^{17} + 39q^{18} - q^{19} - q^{20} - 4q^{21} + 4q^{22} + 29q^{23} + 8q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 36. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{36} - 20x^{35} + 60x^{34} + 1316x^{33} - 9714x^{32} - 23294x^{31} + 401970x^{30} - 317203x^{29} - 8325606x^{28} + 19649023x^{27} + 99800890x^{26} - 371471804x^{25} - 709760653x^{24} + 4044364450x^{23} + 2647919271x^{22} - 28882234242x^{21} - 346001238x^{20} + 142608889677x^{19} - 48711964474x^{18} - 499141043025x^{17} + 258762725221x^{16} + 1251121171888x^{15} - 703058274250x^{14} - 2243311897247x^{13} + 1102242869087x^{12} + 2833426822282x^{11} - 922056603112x^{10} - 2417127378592x^{9} + 235793570629x^{8} + 1257427593087x^{7} + 185545354316x^{6} - 302120342853x^{5} - 116814392666x^{4} + 1797487715x^{3} + 5545435697x^{2} + 270113574x - 71144036$

Atkin–Lehner signs

$p$	$w_p$
727	$-1$

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	6	6	1	9	16	63	−1	29	41	−9	18	43	12	23	77	17	4	−12	−2	17	−14	8	39	13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.239 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, 727]: [0, 61] (certified at ell = 2147482591)
T = [(2, 1), (3, 1)]; h of degree 60 = 1 x 60 (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(727, 2)[1]   # 727.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 727:2 1000