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

Newform orbit 111.6.a.b

Trace form

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

$6q - 9q^{2} - 54q^{3} + 53q^{4} - 58q^{5} + 81q^{6} + 176q^{7} - 327q^{8} + 486q^{9} + 1168q^{10} - 390q^{11} - 477q^{12} + 234q^{13} - 479q^{14} + 522q^{15} + 177q^{16} - 910q^{17} - 729q^{18} + 632q^{19} - 5960q^{20} - 1584q^{21} - 2809q^{22} - 2192q^{23} + 2943q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{6} + 67x^{5} - 5984x^{4} - 147812x^{3} + 15676940x^{2} - 319365284x + 1938558872$

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 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
$\operatorname{tr} a_p$	−9	−54	−58	176	−390	234	−910	632	−2192	−9766	−14152	−8214	−26220	15332	−5172	4180	−37118	5652	28464	−2412	−28754	−25392	−144826	−302474	182332

$\operatorname{tr} a_n$ 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)[1]   # 111.6.a.b

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

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