Atlas › Modular forms › Weight 2 › Level 545 › 545.2.a › 545.2.a.f

Newform orbit 545.2.a.f

Trace form

The 13 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{13} - 110x^{12} + 2253x^{11} + 127438x^{10} - 5083832x^{9} - 13648453x^{8} + 2591591050x^{7} - 21510959312x^{6} - 342591453507x^{5} + 5131062976508x^{4} - 9788850843627x^{3} - 101054305189210x^{2} + 368809465603326x - 169626754112977$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
109	$+1$

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	9	13	16	0	6	5	1	21	−5	11	−2	−12	27	28	3	−28	−8	28	−14	−1	46	41	−20	−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 0.233 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, 5, 109, 545]: [0, 1, 9, 56] (certified at ell = 2147482801)
T = [(2, 1), (3, 3), (7, 5), (11, 7)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 8 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(545, 2)[5]   # 545.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 545:2 1000