Atlas › Modular forms › Weight 2 › Level 577 › 577.2.a › 577.2.a.d

Newform orbit 577.2.a.d

Trace form

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

$18q + 8q^{2} + 24q^{4} - 2q^{5} - 11q^{6} + 11q^{7} + 24q^{8} + 18q^{9} - 7q^{10} + 11q^{11} + 8q^{12} + q^{13} + 19q^{14} + 21q^{15} + 12q^{16} + 14q^{17} + 31q^{18} - 4q^{19} - 6q^{20} + 3q^{21} + 3q^{22} + 47q^{23} - 2q^{24} + O(q^{25})$

Coefficient field

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

$x^{18} - 8x^{17} - 23x^{16} + 274x^{15} + 165x^{14} - 3683x^{13} - 783x^{12} + 24958x^{11} + 7586x^{10} - 86624x^{9} - 44341x^{8} + 135462x^{7} + 80836x^{6} - 92227x^{5} - 46234x^{4} + 31162x^{3} + 6992x^{2} - 4834x + 537$

Atkin–Lehner signs

$p$	$w_p$
577	$-1$

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
$\operatorname{tr} a_p$	8	0	−2	11	11	1	14	−4	47	11	−8	21	6	0	54	−19	53	8	−28	24	−47	13	82	57	−7

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.204 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, 577]: [0, 48] (certified at ell = 2147481793)
T = [(2, 1), (3, 1)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 5 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(577, 2)[3]   # 577.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 577:2 1000