Atlas › Modular forms › Weight 2 › Level 323 › 323.2.a › 323.2.a.e

Newform orbit 323.2.a.e

Trace form

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

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} + x^{5} - 11x^{4} - 7x^{3} + 31x^{2} + 14x - 17$

Atkin–Lehner signs

$p$	$w_p$
17	$+1$
19	$-1$

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	−3	−1	5	2	14	−6	6	−1	−16	9	1	−7	21	17	−21	−6	31	−5	−21	21	−11	8	−13	21

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.182 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, 17, 19, 323]: [0, 2, 2, 32] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 3 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(323, 2)[4]   # 323.2.a.e

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

cargo run --release -p sagebrush-web --example atlas -- 323:2 1000