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

Newform orbit 111.6.a.d

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 9q^{2} + 90q^{3} + 225q^{4} + 36q^{5} + 81q^{6} + 372q^{7} + 81q^{8} + 810q^{9} + 1416q^{10} + 1446q^{11} + 2025q^{12} + 910q^{13} + 591q^{14} + 324q^{15} + 5957q^{16} + 960q^{17} + 729q^{18} + 3520q^{19} - 4146q^{20} + 3348q^{21} - 301q^{22} - 2514q^{23} + 729q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{10} - 45x^{9} - 22504x^{8} + 980598x^{7} + 166248617x^{6} - 6396565335x^{5} - 466754022884x^{4} + 12404835142572x^{3} + 486689462409316x^{2} - 3981043549079628x - 91888937281465720$

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
$\operatorname{tr} a_p$	9	90	36	372	1446	910	960	3520	−2514	9072	13452	−13690	34560	38496	48744	−35988	23868	−44852	93020	−11280	81850	−10952	−253770	−251112	−28536

$\operatorname{tr} a_n$ 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)[3]   # 111.6.a.d

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

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