Atlas › Modular forms › Weight 2 › Level 111 › 111.2.a › 111.2.a.a

Newform orbit 111.2.a.a

Trace form

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

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

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

$x^{3} - 7x^{2} + 13x - 5$

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	−3	4	−4	4	−2	4	−8	−2	16	−8	3	18	−12	−4	−6	6	−6	−16	12	−6	12	0	−4	−14

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.083 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 3, 37, 111]: [0, 1, 3, 14] (certified at ell = 2147483629)
T = [(2, 1), (5, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 1 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, 2)[0]   # 111.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 111:2 1000