Atlas › Modular forms › Weight 6 › Level 94 › 94.6.a › 94.6.a.d

Newform orbit 94.6.a.d

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q + 24q^{2} + 14q^{3} + 96q^{4} + 82q^{5} + 56q^{6} + 88q^{7} + 384q^{8} + 856q^{9} + 328q^{10} + 644q^{11} + 224q^{12} + 1324q^{13} + 352q^{14} + 3706q^{15} + 1536q^{16} + 1630q^{17} + 3424q^{18} + 644q^{19} + 1312q^{20} - 1356q^{21} + 2576q^{22} + 4736q^{23} + 896q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{6} - 96x^{5} - 13793x^{4} + 1578592x^{3} + 1611333x^{2} - 2326917702x - 14535472860$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
47	$+1$

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
$\operatorname{tr} a_p$	24	14	82	88	644	1324	1630	644	4736	6580	4538	−2554	−25774	20824	−13254	25078	9232	−70754	11990	−58310	−133324	3834	−17336	−106786	−156162

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.265 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, 47, 94]: [1, 2, 21, 62] (certified at ell = 2147483137)
T = [(3, 1), (5, 1)]; h of degree 20 = 1 x 20 (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(94, 6)[3]   # 94.6.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 94:6 1000