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

Newform orbit 109.6.a.b

Trace form

The 24 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$24q + 12q^{2} + 44q^{3} + 432q^{4} + 182q^{5} + 349q^{6} + 99q^{7} + 363q^{8} + 2428q^{9} + 436q^{10} + 2701q^{11} + 1020q^{12} + 1107q^{13} + 2507q^{14} + 1656q^{15} + 6792q^{16} + 2047q^{17} + 1472q^{18} + 4845q^{19} + 8253q^{20} + 757q^{21} - 549q^{22} + 6933q^{23} + 17125q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 24. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{24} - 56x^{23} - 3511x^{22} + 236226x^{21} + 4328953x^{20} - 403005404x^{19} - 2044086111x^{18} + 372889654714x^{17} - 157251095636x^{16} - 210548840782972x^{15} + 582880923891653x^{14} + 76575860864381464x^{13} - 265632375138438416x^{12} - 18246472991531084388x^{11} + 58220224555327234611x^{10} + 2809862917492167539472x^{9} - 6576189382691140382800x^{8} - 266541616181812758583064x^{7} + 331703699452029390126896x^{6} + 14139224350257811346716160x^{5} - 3139065750762623138621184x^{4} - 350611611715677291444124416x^{3} - 102170448036277400403903744x^{2} + 3164003897449722928352100864x + 1398147460711239957746162688$

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 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
$\operatorname{tr} a_p$	12	44	182	99	2701	1107	2047	4845	6933	16478	31869	11910	39000	16969	66384	10487	224515	−28658	−86309	129506	21258	−151072	107379	167123	−213538

$\operatorname{tr} a_n$ 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)[1]   # 109.6.a.b

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

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