Atlas › Modular forms › Weight 2 › Level 109 › 109.2.a › 109.2.a.b

Newform orbit 109.2.a.b

Trace form

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

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

Coefficient field

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

$x^{3} + 14x^{2} + 56x + 56$

Atkin–Lehner signs

$p$	$w_p$
109	$+1$

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
$\operatorname{tr} a_p$	−2	−4	−6	−1	−13	−1	3	−5	1	−6	−7	0	−6	9	−10	9	−25	−10	−11	−10	20	−6	−13	−21	−20

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.16 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, 109]: [0, 9] (certified at ell = 2147482801)
T = [(2, 1), (3, 3)]; h of degree 8 = 1 x 8 (phi(ord eps) x dim S_k^new); 2 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(109, 2)[1]   # 109.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 109:2 1000