Atlas › Modular forms › Weight 4 › Level 58 › 58.4.a › 58.4.a.d

Newform orbit 58.4.a.d

Trace form

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

$3q + 6q^{2} + 2q^{3} + 12q^{4} + 20q^{5} + 4q^{6} + 24q^{7} + 24q^{8} + 5q^{9} + 40q^{10} + 10q^{11} + 8q^{12} - 4q^{13} + 48q^{14} - 130q^{15} + 48q^{16} - 66q^{17} + 10q^{18} - 164q^{19} + 80q^{20} - 88q^{21} + 20q^{22} - 204q^{23} + 16q^{24} + O(q^{25})$

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

$x^{3} - 22x^{2} + 102x + 108$

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
$\operatorname{tr} a_p$	6	2	20	24	10	−4	−66	−164	−204	87	−86	−42	562	18	654	712	184	322	−228	−52	−494	−2110	−288	914	218

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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, 2, 9, 25] (certified at ell = 2147483549)
T = [(3, 1), (5, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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, 4)[3]   # 58.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 58:4 1000