Atlas › Modular forms › Weight 2 › Level 818 › 818.2.a › 818.2.a.a

Newform orbit 818.2.a.a

Trace form

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

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

Coefficient field

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

$x^{4} + 7x^{3} + 15x^{2} + 9x - 1$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
409	$-1$

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	−2	−5	−7	−9	−8	−7	−5	2	−8	5	−20	2	−16	16	−6	3	−4	−13	11	−4	3	−2	3	−5

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.246 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, 409, 818]: [0, 1, 34, 104] (certified at ell = 2147477809)
T = [(3, 1), (5, 1)]; h of degree 35 = 1 x 35 (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(818, 2)[0]   # 818.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 818:2 1000