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

Newform orbit 89.4.a.c

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$6q - q^{3} + 6q^{4} - 27q^{5} - 71q^{6} - 82q^{7} - 72q^{8} + 3q^{9} + 25q^{10} + 10q^{11} + 61q^{12} - 114q^{13} - 8q^{14} - 203q^{15} - 154q^{16} - 63q^{17} - 75q^{18} - 213q^{19} - 171q^{20} - 96q^{21} - 174q^{22} + 55q^{23} + 111q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} + x^{5} - 38x^{4} - 62x^{3} + 356x^{2} + 764x + 22$

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 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
$\operatorname{tr} a_p$	0	−1	−27	−82	10	−114	−63	−213	55	−360	−467	440	−432	−733	280	−1073	398	−150	−1160	−206	801	−2646	−1410	534	1195

$\operatorname{tr} a_n$ 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)[2]   # 89.4.a.c

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

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