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

Newform orbit 58.6.a.d

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q + 16q^{2} + 13q^{3} + 64q^{4} + 93q^{5} + 52q^{6} + 134q^{7} + 256q^{8} + 779q^{9} + 372q^{10} + 123q^{11} + 208q^{12} + 851q^{13} + 536q^{14} + 1075q^{15} + 1024q^{16} + 2502q^{17} + 3116q^{18} + 26q^{19} + 1488q^{20} - 926q^{21} + 492q^{22} - 924q^{23} + 832q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{4} - 106x^{3} - 2496x^{2} + 100560x + 100224$

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
$\operatorname{tr} a_p$	16	13	93	134	123	851	2502	26	−924	−3364	5999	−18454	−3492	−853	−11997	2973	−32100	41486	12920	67866	173798	70745	47208	172992	−15832

$\operatorname{tr} a_n$ 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)[3]   # 58.6.a.d

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

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