Atlas › Modular forms › Weight 6 › Level 111 › 111.6.a › 111.6.a.c

Newform orbit 111.6.a.c

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$9q + 3q^{2} - 81q^{3} + 133q^{4} + 42q^{5} - 27q^{6} - 20q^{7} + 249q^{8} + 729q^{9} - 468q^{10} - 498q^{11} - 1197q^{12} + 44q^{13} + 543q^{14} - 378q^{15} + 3085q^{16} + 3354q^{17} + 243q^{18} + 704q^{19} + 7962q^{20} + 180q^{21} + 4049q^{22} + 6222q^{23} - 2241q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{9} - 45x^{8} - 15980x^{7} + 437466x^{6} + 85744671x^{5} - 1214042481x^{4} - 154228330600x^{3} + 2010156194988x^{2} + 74276944596324x - 1140432891935112$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
37	$-1$

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
$\operatorname{tr} a_p$	3	−81	42	−20	−498	44	3354	704	6222	20118	6872	12321	9366	−32904	−22296	36462	−840	12630	−11984	115800	−81744	−27756	−58098	301650	−10

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.28 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, 37, 111]: [1, 3, 17, 66] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 7 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(111, 6)[2]   # 111.6.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 111:6 1000