Atlas › Modular forms › Weight 6 › Level 58 › 58.6.a › 58.6.a.b

Newform orbit 58.6.a.b

Trace form

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

$3q - 12q^{2} + 14q^{3} + 48q^{4} + 68q^{5} - 56q^{6} + 24q^{7} - 192q^{8} + 61q^{9} - 272q^{10} + 102q^{11} + 224q^{12} + 220q^{13} - 96q^{14} + 2154q^{15} + 768q^{16} + 3470q^{17} - 244q^{18} + 3652q^{19} + 1088q^{20} + 4440q^{21} - 408q^{22} + 5676q^{23} - 896q^{24} + O(q^{25})$

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

$x^{3} - 82x^{2} - 4766x + 384036$

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
$\operatorname{tr} a_p$	−12	14	68	24	102	220	3470	3652	5676	2523	−1890	−3794	−6862	−21570	3866	25848	27976	−22710	−38956	62116	−50014	−62346	151424	−33742	−151254

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.241 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, 13, 39] (certified at ell = 2147483549)
T = [(3, 1), (5, 1)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 4 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, 6)[1]   # 58.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 58:6 1000