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

Newform orbit 58.8.a.c

Trace form

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

$4q - 32q^{2} - 68q^{3} + 256q^{4} - 180q^{5} + 544q^{6} - 576q^{7} - 2048q^{8} + 5648q^{9} + 1440q^{10} + 5388q^{11} - 4352q^{12} - 3396q^{13} + 4608q^{14} + 42820q^{15} + 16384q^{16} + 4712q^{17} - 45184q^{18} + 11384q^{19} - 11520q^{20} - 179648q^{21} - 43104q^{22} - 197848q^{23} + 34816q^{24} + O(q^{25})$

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

$x^{4} + 248x^{3} - 142068x^{2} - 7826832x - 98075340$

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
$\operatorname{tr} a_p$	−32	−68	−180	−576	5388	−3396	4712	11384	−197848	97556	−513692	−224688	−374176	407196	−1284708	360068	−2535976	4348968	34848	−2191368	−5406640	1270852	−21683592	1789280	−372576

$\operatorname{tr} a_n$ 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)[2]   # 58.8.a.c

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

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