Atlas › Modular forms › Weight 2 › Level 318 › 318.2.a › 318.2.a.g

Newform orbit 318.2.a.g

Trace form

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

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

Coefficient field

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

$x^{2} - 4x - 13$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
3	$-1$
53	$-1$

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	1	1	−2	−2	−5	3	0	−7	1	2	−13	3	0	2	−3	5	9	−5	0	15	−18	−1	0

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.331 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, 53, 106, 159, 318]: [0, 1, 1, 3, 5, 15, 20, 58] (certified at ell = 2147483053)
T = [(5, 1), (7, 3)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 2 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(318, 2)[6]   # 318.2.a.g

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

cargo run --release -p sagebrush-web --example atlas -- 318:2 1000