Atlas › Modular forms › Weight 6 › Level 93 › 93.6.a › 93.6.a.b

Newform orbit 93.6.a.b

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 9q^{2} - 45q^{3} + 53q^{4} + 36q^{5} - 81q^{6} + 108q^{7} - 153q^{8} + 405q^{9} - 247q^{10} + 876q^{11} - 477q^{12} + 880q^{13} + 1164q^{14} - 324q^{15} + 913q^{16} + 1962q^{17} + 729q^{18} + 2420q^{19} + 5439q^{20} - 972q^{21} + 6355q^{22} + 6990q^{23} + 1377q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{5} - 45x^{4} - 8887x^{3} + 230877x^{2} + 11151382x - 62206416$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
31	$-1$

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
$\operatorname{tr} a_p$	9	−45	36	108	876	880	1962	2420	6990	4704	4805	−2940	−7158	−650	20460	15402	33102	26174	41272	−14550	−94452	−94854	201390	150150	68974

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 3, 31, 93]: [1, 3, 15, 56] (certified at ell = 2147482951)
T = [(2, 1), (5, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 6 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(93, 6)[1]   # 93.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 93:6 1000