Atlas › Modular forms › Weight 4 › Level 134 › 134.4.a › 134.4.a.c

Newform orbit 134.4.a.c

Trace form

The 3 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$3q - 6q^{2} - 3q^{3} + 12q^{4} - 6q^{5} + 6q^{6} - 5q^{7} - 24q^{8} + 18q^{9} + 12q^{10} - 12q^{12} - 81q^{13} + 10q^{14} - 27q^{15} + 48q^{16} + 13q^{17} - 36q^{18} - 234q^{19} - 24q^{20} - 314q^{21} + 75q^{23} + 24q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 9x^{2} - x - 8$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
67	$-1$

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
$\operatorname{tr} a_p$	−6	−3	−6	−5	0	−81	13	−234	75	−243	−532	−34	107	131	572	−186	591	−813	201	402	−391	−485	1761	−1349	−1736

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.111 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 engine's checks

dim M^+ at levels [1, 2, 67, 134]: [1, 2, 18, 53] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 17 = 1 x 17 (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(134, 4)[2]   # 134.4.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 134:4 1000