Atlas › Modular forms › Weight 6 › Level 107 › 107.6.a › 107.6.a.a

Newform orbit 107.6.a.a

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q - 8q^{2} - 35q^{3} + 224q^{4} - 201q^{5} - 87q^{6} - 404q^{7} - 597q^{8} + 1194q^{9} - 368q^{10} - 380q^{11} - 1908q^{12} - 3880q^{13} - 1249q^{14} - 1741q^{15} - 252q^{16} - 2327q^{17} - 6144q^{18} - 846q^{19} - 5127q^{20} - 5005q^{21} - 6995q^{22} - 5618q^{23} - 7789q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{19} + 43x^{18} - 2310x^{17} - 101667x^{16} + 2221662x^{15} + 92428092x^{14} - 1244953876x^{13} - 41065545751x^{12} + 464761450750x^{11} + 9333764242895x^{10} - 112672776396412x^{9} - 1005395334566394x^{8} + 15676730195601239x^{7} + 28612674570825566x^{6} - 1069157144017244862x^{5} + 2587972057315470640x^{4} + 24532435833177990592x^{3} - 144978661810715063632x^{2} + 230485485966522794176x - 38430943683093522944$

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 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
$\operatorname{tr} a_p$	−8	−35	−201	−404	−380	−3880	−2327	−846	−5618	−17171	−26592	−38723	−7696	−34651	14457	−15054	5615	−97415	−44330	60305	−196030	137037	129394	−68806	−413400

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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, 107]: [1, 46] (certified at ell = 2147482621)
T = [(2, 1), (3, 1)]; h of degree 44 = 1 x 44 (phi(ord eps) x dim S_k^new); 8 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, 6)[0]   # 107.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 107:6 1000