Atlas › Modular forms › Weight 2 › Level 498 › 498.2.a › 498.2.a.f

Newform orbit 498.2.a.f

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 3. It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{3} + 126x^{2} + 3264x - 19744$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
3	$+1$
83	$+1$

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
$\operatorname{tr} a_p$	−3	−3	0	−3	−3	−3	−9	6	−18	−12	−9	−9	−3	−3	−24	0	−3	3	0	−6	−12	0	−3	9	6

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.385 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, 3, 6, 83, 166, 249, 498]: [0, 1, 1, 3, 8, 23, 30, 88] (certified at ell = 2147483323)
T = [(5, 1), (7, 3), (11, 5), (13, 7), (17, 9)]; h of degree 15 = 1 x 15 (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(498, 2)[5]   # 498.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 498:2 1000