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

Newform orbit 727.2.a.a

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q - 13q^{2} - 8q^{3} + 21q^{4} - 8q^{5} - 3q^{6} - 7q^{7} - 36q^{8} + 2q^{9} - 4q^{10} - 13q^{11} - 16q^{12} - 18q^{13} - 8q^{14} - 16q^{15} + 23q^{16} - 55q^{17} - 12q^{18} - 3q^{19} - 9q^{20} - 20q^{21} - 6q^{22} - 27q^{23} + 14q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{24} + 21x^{23} + 152x^{22} + 240x^{21} - 2346x^{20} - 11214x^{19} + 540x^{18} + 102005x^{17} + 153597x^{16} - 360545x^{15} - 1040209x^{14} + 264958x^{13} + 2955777x^{12} + 1554402x^{11} - 3742820x^{10} - 4195891x^{9} + 1144272x^{8} + 3617149x^{7} + 1316949x^{6} - 662407x^{5} - 639589x^{4} - 180440x^{3} - 18165x^{2} - 98x + 4$

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 odd 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
$\operatorname{tr} a_p$	−13	−8	−8	−7	−13	−18	−55	−3	−27	−45	−1	−12	−41	−24	−25	−61	7	10	−4	−18	−25	−6	−38	−65	−21

$\operatorname{tr} a_n$ 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)[0]   # 727.2.a.a

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

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