Atlas › Modular forms › Weight 2 › Level 107 › 107.2.a › 107.2.a.b

Newform orbit 107.2.a.b

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - q^{2} + 3q^{3} + 7q^{4} + 5q^{5} - 5q^{6} + 4q^{7} - 6q^{8} + 6q^{9} - q^{10} - 2q^{11} + 6q^{12} + 18q^{13} - 9q^{14} - 9q^{15} - q^{16} - q^{17} - 17q^{18} - 4q^{19} - 10q^{20} - 11q^{21} - 5q^{22} - 31q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + 3T_{3}$ , whose minimal polynomial is

$x^{7} - 8x^{6} - 85x^{5} + 809x^{4} + 986x^{3} - 17538x^{2} + 2188x + 111064$

Atkin–Lehner signs

$p$	$w_p$
107	$-1$

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
$\operatorname{tr} a_p$	−1	3	5	4	−2	18	−1	−4	0	−3	4	25	0	11	−9	8	−19	25	−24	−19	30	−21	12	−22	−4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.159 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, 107]: [0, 10] (certified at ell = 2147482621)
T = [(2, 1), (3, 3)]; h of degree 9 = 1 x 9 (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(107, 2)[1]   # 107.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 107:2 1000