Atlas › Modular forms › Weight 2 › Level 898 › 898.2.a › 898.2.a.f

Newform orbit 898.2.a.f

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{3} - 8x^{2} + 17x - 8$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
449	$-1$

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	2	6	6	−1	−1	10	8	6	6	−8	−9	−3	−12	19	10	21	−2	−5	−13	−10	39	−2	−3	3

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.377 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, 449, 898]: [0, 1, 38, 114] (certified at ell = 2147479937)
T = [(3, 1), (5, 1)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 4 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(898, 2)[5]   # 898.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 898:2 1000