Atlas › Modular forms › Weight 2 › Level 321 › 321.2.a › 321.2.a.a

Newform orbit 321.2.a.a

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{5})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{2} - x - 1$

Atkin–Lehner signs

$p$	$w_p$
3	$+1$
107	$+1$

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
$\operatorname{tr} a_p$	−1	−2	2	−4	−6	−2	2	−8	−4	−2	−12	2	0	−6	6	14	8	2	10	−8	2	−4	−12	−10	−2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.185 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, 3, 107, 321]: [0, 1, 10, 38] (certified at ell = 2147482621)
T = [(2, 1), (5, 1)]; h of degree 17 = 1 x 17 (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(321, 2)[0]   # 321.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 321:2 1000