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

Newform orbit 321.2.a.d

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q - 7q^{3} + 14q^{4} - 8q^{5} + 6q^{7} + 3q^{8} + 7q^{9} + 7q^{10} + 4q^{11} - 14q^{12} + 6q^{13} + 12q^{14} + 8q^{15} + 32q^{16} - 10q^{17} + 8q^{19} - 17q^{20} - 6q^{21} + 10q^{22} + 6q^{23} - 3q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{7} + 8x^{6} - 15x^{5} - 191x^{4} - 133x^{3} + 613x^{2} + 664x + 167$

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
$\operatorname{tr} a_p$	0	−7	−8	6	4	6	−10	8	6	0	16	10	−2	2	16	−16	20	2	30	32	−12	36	−10	−4	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)[3]   # 321.2.a.d

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

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