Atlas › Modular forms › Weight 2 › Level 890 › 890.2.a › 890.2.a.n

Newform orbit 890.2.a.n

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 2T_{7} + 3T_{17} + 4T_{19} + 5T_{23} + 6T_{11} + 7T_{13} + 8T_{29}$ , whose minimal polynomial is

$x^{6} - 97x^{5} - 1521x^{4} + 263913x^{3} + 1440846x^{2} - 199484612x - 2300296392$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
5	$-1$
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
$\operatorname{tr} a_p$	−6	3	6	6	9	6	−6	10	−4	−2	16	4	4	−5	5	−17	6	5	20	26	−18	4	7	−6	34

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.956 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, 2, 5, 10, 89, 178, 445, 890]: [0, 1, 1, 3, 8, 24, 46, 138] (certified at ell = 2147482921)
T = [(3, 1), (7, 2), (17, 3), (19, 4), (23, 5), (11, 6), (13, 7), (29, 8)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 9 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(890, 2)[13]   # 890.2.a.n

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

cargo run --release -p sagebrush-web --example atlas -- 890:2 1000