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

Newform orbit 321.2.a.c

Trace form

The 6 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{6} - 9x^{5} + 22x^{4} - 5x^{3} - 40x^{2} + 41x - 9$

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 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
$\operatorname{tr} a_p$	3	6	6	0	6	−8	4	−4	14	10	12	−12	−6	−12	16	12	8	−24	−4	36	−26	8	−8	−8	−24

$\operatorname{tr} a_n$ 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)[2]   # 321.2.a.c

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

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