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

Newform orbit 58.8.a.b

Trace form

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

$3q + 24q^{2} - 25q^{3} + 192q^{4} - 305q^{5} - 200q^{6} - 1138q^{7} + 1536q^{8} - 1862q^{9} - 2440q^{10} - 7227q^{11} - 1600q^{12} - 14087q^{13} - 9104q^{14} - 13605q^{15} + 12288q^{16} - 12720q^{17} - 14896q^{18} - 31528q^{19} - 19520q^{20} - 70842q^{21} - 57816q^{22} - 57316q^{23} - 12800q^{24} + O(q^{25})$

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

$x^{3} + 330x^{2} + 5508x + 23296$

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 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
$\operatorname{tr} a_p$	24	−25	−305	−1138	−7227	−14087	−12720	−31528	−57316	−73167	−87623	57064	−84210	−121747	−341039	−305509	274600	3573584	8500452	748250	5239756	2735659	−3727844	−14594358	−6298990

$\operatorname{tr} a_n$ 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)[1]   # 58.8.a.b

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

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