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

Newform orbit 93.6.a.a

Trace form

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

$4q - 3q^{2} + 36q^{3} + 49q^{4} - 42q^{5} - 27q^{6} - 284q^{7} - 735q^{8} + 324q^{9} - 1075q^{10} - 252q^{11} + 441q^{12} - 1486q^{13} + 1500q^{14} - 378q^{15} + 2049q^{16} - 2664q^{17} - 243q^{18} - 4152q^{19} + 4209q^{20} - 2556q^{21} - 3077q^{22} + 2358q^{23} - 6615q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{4} + 45x^{3} - 4243x^{2} - 32409x + 1703974$

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 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
$\operatorname{tr} a_p$	−3	36	−42	−284	−252	−1486	−2664	−4152	2358	−942	3844	−24722	−684	−36430	−17316	−4164	24582	−41880	16820	50286	−14678	15850	−34314	216012	−286756

$\operatorname{tr} a_n$ 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)[0]   # 93.6.a.a

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

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