Atlas › Modular forms › Weight 4 › Level 89 › 89.4.a › 89.4.a.d

Newform orbit 89.4.a.d

Trace form

The 14 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$14q + 3q^{2} + 6q^{3} + 79q^{4} + 6q^{5} + 15q^{6} + 76q^{7} + 39q^{8} + 164q^{9} + 99q^{10} + 98q^{11} + 111q^{12} + 48q^{13} + 76q^{14} + 144q^{15} + 495q^{16} + 82q^{17} + 220q^{18} + 362q^{19} + 143q^{20} + 260q^{21} + 362q^{22} + 162q^{23} - 569q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 14. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{14} - 9x^{13} - 341x^{12} + 2877x^{11} + 43677x^{10} - 338467x^{9} - 2705301x^{8} + 18711943x^{7} + 86801244x^{6} - 510170688x^{5} - 1404278932x^{4} + 6374836028x^{3} + 9601798072x^{2} - 26943594240x - 10559703312$

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
$\operatorname{tr} a_p$	3	6	6	76	98	48	82	362	162	−48	556	156	−92	856	−342	−314	732	628	922	2788	1982	4636	304	−1246	−1366

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.101 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, 24] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 22 = 1 x 22 (phi(ord eps) x dim S_k^new); 3 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, 4)[3]   # 89.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 89:4 1000