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

Newform orbit 547.2.a.b

Trace form

The 18 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} - 5q^{10} + 2q^{11} - 32q^{12} - 25q^{13} - 7q^{14} + 9q^{15} + 8q^{16} - 30q^{17} - 10q^{18} + 4q^{19} - 41q^{20} - 16q^{21} - 24q^{22} - 26q^{23} - 12q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 18. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{18} + 34x^{17} + 255x^{16} - 3195x^{15} - 48994x^{14} + 16392x^{13} + 2781279x^{12} + 6366898x^{11} - 72100121x^{10} - 236853079x^{9} + 954118420x^{8} + 3304529852x^{7} - 6517466429x^{6} - 18660492525x^{5} + 18407238076x^{4} + 36746402496x^{3} - 8254246851x^{2} - 5051748480x - 432751568$

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 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
$\operatorname{tr} a_p$	−4	−10	−27	−11	2	−25	−30	4	−26	−18	−5	−18	−17	8	−52	−60	−8	−26	12	−1	−2	18	−43	−28	−34

$\operatorname{tr} a_n$ 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)[1]   # 547.2.a.b

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

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