Atlas › Modular forms › Weight 2 › Level 329 › 329.2.a › 329.2.a.f

Newform orbit 329.2.a.f

Trace form

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

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

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

$x^{5} - 3x^{4} - 25x^{3} + 76x^{2} + 38x - 83$

Atkin–Lehner signs

$p$	$w_p$
7	$+1$
47	$-1$

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
$\operatorname{tr} a_p$	1	2	−4	−5	4	1	−3	−2	−7	4	10	4	−19	14	5	34	−7	3	−28	27	−51	47	1	1	−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 0.185 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, 7, 47, 329]: [0, 1, 5, 34] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 23 = 1 x 23 (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(329, 2)[5]   # 329.2.a.f

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

cargo run --release -p sagebrush-web --example atlas -- 329:2 1000