Atlas › Modular forms › Weight 2 › Level 489 › 489.2.a › 489.2.a.d

Newform orbit 489.2.a.d

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{10} - 42x^{8} + 14x^{7} + 595x^{6} - 281x^{5} - 3518x^{4} + 1821x^{3} + 8375x^{2} - 3659x - 5768$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
163	$+1$

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	10	−1	14	4	16	−3	13	−1	−5	11	22	−15	14	0	−7	−5	23	10	−1	23	14	−4	−5	−2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.198 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, 3, 163, 489]: [0, 1, 14, 56] (certified at ell = 2147483179)
T = [(2, 1), (5, 1)]; h of degree 27 = 1 x 27 (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(489, 2)[3]   # 489.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 489:2 1000