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

Newform orbit 323.2.a.d

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{5} + 6x^{4} + x^{3} - 22x^{2} - 15x + 1$

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 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
$\operatorname{tr} a_p$	−3	−3	−3	−5	0	−20	−5	−5	5	6	−3	−19	5	5	9	−3	6	−23	−3	21	−13	−3	−16	−49	−13

$\operatorname{tr} a_n$ 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)[3]   # 323.2.a.d

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

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