Atlas › Modular forms › Weight 2 › Level 327 › 327.2.a › 327.2.a.d

Newform orbit 327.2.a.d

Trace form

The 9 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

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

Coefficient field

The coefficients $a_n$ generate a number field of degree 9. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{9} - 4x^{8} - 36x^{7} + 144x^{6} + 338x^{5} - 1438x^{4} - 1079x^{3} + 5202x^{2} + 964x - 5866$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
109	$+1$

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	9	1	6	−5	6	0	13	−5	−3	18	16	2	4	3	0	−22	−3	14	−26	17	−22	−14	−5	27

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.186 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, 109, 327]: [0, 1, 9, 38] (certified at ell = 2147482801)
T = [(2, 1), (5, 1)]; h of degree 19 = 1 x 19 (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(327, 2)[3]   # 327.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 327:2 1000