Atlas › Modular forms › Weight 4 › Level 111 › 111.4.a › 111.4.a.d

Newform orbit 111.4.a.d

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 3q^{2} - 9q^{3} + 9q^{4} - 6q^{5} + 9q^{6} + 15q^{7} - 15q^{8} + 27q^{9} - 24q^{10} - 81q^{11} - 27q^{12} - 21q^{13} - 105q^{14} + 18q^{15} - 87q^{16} - 165q^{17} - 27q^{18} + 21q^{19} - 168q^{20} - 45q^{21} - 9q^{22} - 99q^{23} + 45q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{3} + 9x^{2} - 108x - 918$

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
$\operatorname{tr} a_p$	−3	−9	−6	15	−81	−21	−165	21	−99	−306	54	111	−366	360	−342	−543	−66	66	−30	−1248	225	−1254	627	−1245	−198

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.111 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, 2, 11, 40] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 3 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, 4)[3]   # 111.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 111:4 1000