Atlas › Modular forms › Weight 2 › Level 547 › 547.2.a › 547.2.a.c

Newform orbit 547.2.a.c

Trace form

The 25 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$25q + 4q^{2} + 8q^{3} + 26q^{4} + 29q^{5} + q^{6} + 5q^{7} + 6q^{8} + 29q^{9} - q^{10} + 10q^{11} + 14q^{12} + 19q^{13} + 9q^{14} + 5q^{15} + 16q^{16} + 40q^{17} - 8q^{18} + 33q^{20} - 8q^{21} - 10q^{22} + 26q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 25. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{25} - 28x^{24} - 117x^{23} + 9929x^{22} - 43855x^{21} - 1363320x^{20} + 11770074x^{19} + 86703929x^{18} - 1218143637x^{17} - 1792974807x^{16} + 67806194283x^{15} - 90000888780x^{14} - 2165200388084x^{13} + 7027320006180x^{12} + 37744288454364x^{11} - 202435650401052x^{10} - 255100535157612x^{9} + 2943567871498945x^{8} - 1822501327101462x^{7} - 20009435452773222x^{6} + 37070568848220987x^{5} + 35012432638213166x^{4} - 140558734364355690x^{3} + 94764580560059956x^{2} + 6379192437615424x - 4100708899848448$

Atkin–Lehner signs

$p$	$w_p$
547	$-1$

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
$\operatorname{tr} a_p$	4	8	29	5	10	19	40	0	26	30	−5	26	9	−10	28	80	−2	22	−16	−1	2	−34	15	38	−2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.208 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, 547]: [0, 46] (certified at ell = 2147481883)
T = [(2, 1), (3, 3)]; h of degree 45 = 1 x 45 (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(547, 2)[2]   # 547.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 547:2 1000