Atlas › Modular forms › Weight 2 › Level 998 › 998.2.a › 998.2.a.f

Newform orbit 998.2.a.f

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

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

$x^{4} + 14x^{3} - 683x^{2} - 3981x + 25209$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
499	$+1$

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
$\operatorname{tr} a_p$	−4	−1	−5	−8	6	−3	12	0	−7	−19	−12	−14	5	4	0	1	9	−34	1	16	−8	3	−13	21	−43

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.502 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, 499, 998]: [0, 1, 42, 127] (certified at ell = 2147475601)
T = [(3, 1), (5, 3), (7, 5), (11, 7)]; h of degree 42 = 1 x 42 (phi(ord eps) x dim S_k^new); 10 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(998, 2)[5]   # 998.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 998:2 1000