Atlas › Modular forms › Weight 6 › Level 89 › 89.6.a › 89.6.a.b

Newform orbit 89.6.a.b

Trace form

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

$21q + 7q^{2} + 27q^{3} + 415q^{4} + 61q^{5} + 183q^{6} + 538q^{7} + 351q^{8} + 2216q^{9} + 35q^{10} + 486q^{11} + 759q^{12} + 1588q^{13} + 1116q^{14} + 3441q^{15} + 10839q^{16} + 177q^{17} + 1936q^{18} + 6911q^{19} + 2687q^{20} + 2288q^{21} + 2386q^{22} + 4679q^{23} + 38941q^{24} + O(q^{25})$

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

$x^{21} - 34x^{20} - 3808x^{19} + 126724x^{18} + 6066167x^{17} - 193405642x^{16} - 5404488644x^{15} + 159435841626x^{14} + 3009782939429x^{13} - 78024719409244x^{12} - 1100195730707100x^{11} + 23272450677839552x^{10} + 266908972726115013x^{9} - 4139201325112110578x^{8} - 41765366628345153816x^{7} + 402089799120410376858x^{6} + 3836861955173377184818x^{5} - 16350901293463902120726x^{4} - 159421954024603309843140x^{3} - 1085778519425386137184x^{2} + 459773424308937175299840x - 68658727515398122215168$

Atkin–Lehner signs

$p$	$w_p$
89	$-1$

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
$\operatorname{tr} a_p$	7	27	61	538	486	1588	177	6911	4679	17162	32237	13650	10622	43767	−29464	−26941	−11318	106424	42812	21562	−65103	242554	54146	166341	463479

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.252 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, 89]: [1, 38] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 36 = 1 x 36 (phi(ord eps) x dim S_k^new); 7 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(89, 6)[1]   # 89.6.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 89:6 1000