Atlas › Modular forms › Weight 6 › Level 109 › 109.6.a › 109.6.a.a

Newform orbit 109.6.a.a

Trace form

The 21 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$21q - 8q^{2} - 64q^{3} + 288q^{4} - 218q^{5} - 155q^{6} - 293q^{7} - 597q^{8} + 1213q^{9} - 164q^{10} - 1897q^{11} - 2436q^{12} - 245q^{13} - 2197q^{14} - 1944q^{15} + 1416q^{16} - 3155q^{17} - 3388q^{18} - 5985q^{19} - 10947q^{20} - 6299q^{21} - 5389q^{22} - 7879q^{23} - 7067q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 21. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{21} + 72x^{20} - 891x^{19} - 160958x^{18} - 1008373x^{17} + 144727188x^{16} + 1735713959x^{15} - 68291246184x^{14} - 942608162186x^{13} + 18806478611202x^{12} + 241628105742557x^{11} - 3222368801373630x^{10} - 29779318337915178x^{9} + 342877395517225252x^{8} + 1503249438271286683x^{7} - 18975206093165015550x^{6} - 14103577644763518096x^{5} + 412196823511775437220x^{4} - 595701134743005678980x^{3} - 674029858758527773472x^{2} + 214359555563686045520x - 13347600053314858208$

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
$\operatorname{tr} a_p$	−8	−64	−218	−293	−1897	−245	−3155	−5985	−7879	−20026	−20523	−22934	−40806	−5995	−30578	−19955	−133009	6194	22737	−97514	−62	−1958	−264465	−248853	357330

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.278 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, 109]: [1, 47] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 45 = 1 x 45 (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(109, 6)[0]   # 109.6.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 109:6 1000