Atlas › Modular forms › Weight 2 › Level 89 › 89.2.a › 89.2.a.c

Newform orbit 89.2.a.c

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q - q^{2} - 3q^{3} + 11q^{4} - q^{5} - 3q^{6} + 8q^{7} + 3q^{8} + 2q^{9} - 3q^{10} + 6q^{11} - 15q^{12} - 14q^{14} + 9q^{15} + 11q^{16} - 13q^{17} - 18q^{18} + 13q^{19} - 31q^{20} - 12q^{21} - 14q^{22} + q^{23} - 5q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} + 4x^{4} - 8x^{3} - 6x^{2} + 10x - 2$

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
$\operatorname{tr} a_p$	−1	−3	−1	8	6	0	−13	13	1	2	19	−14	−2	1	−4	−11	0	4	4	−2	−25	54	−20	5	13

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.163 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 89]: [0, 8] (certified at ell = 2147482921)
T = [(2, 1), (3, 1)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 1 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, 2)[2]   # 89.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 89:2 1000