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

Newform orbit 498.2.a.c

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{21})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13} + 9T_{17}$ , whose minimal polynomial is

$x^{2} - 76x + 1360$

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 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
$\operatorname{tr} a_p$	−2	2	−3	1	0	7	3	7	9	0	13	−8	3	1	0	21	−18	16	7	−6	−8	4	−2	−12	−20

$\operatorname{tr} a_n$ 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)[2]   # 498.2.a.c

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

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