Atlas › Modular forms › Weight 8 › Level 58 › 58.8.a › 58.8.a.a

Newform orbit 58.8.a.a

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 24q^{2} + 13q^{3} + 192q^{4} + 195q^{5} - 104q^{6} + 110q^{7} - 1536q^{8} + 3826q^{9} - 1560q^{10} + 3471q^{11} + 832q^{12} - 13467q^{13} - 880q^{14} - 44395q^{15} + 12288q^{16} + 10908q^{17} - 30608q^{18} + 99664q^{19} + 12480q^{20} + 6994q^{21} - 27768q^{22} + 84728q^{23} - 6656q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} - 208x^{2} - 135264x - 799848$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
29	$+1$

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
$\operatorname{tr} a_p$	−24	13	195	110	3471	−13467	10908	99664	84728	−73167	−33969	505336	1351474	549895	1779167	1145919	−2870284	−4599148	3643276	3106454	1560220	−3683003	−443064	6888166	−10958826

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.299 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 engine's checks

dim M^+ at levels [1, 2, 29, 58]: [1, 3, 19, 55] (certified at ell = 2147483549)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (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(58, 8)[0]   # 58.8.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 58:8 1000