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

Newform orbit 134.4.a.e

Trace form

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

$6q - 12q^{2} + 24q^{4} + 9q^{5} - 5q^{7} - 48q^{8} + 90q^{9} - 18q^{10} - 11q^{11} + 62q^{13} + 10q^{14} + 33q^{15} + 96q^{16} + 29q^{17} - 180q^{18} + 215q^{19} + 36q^{20} + 632q^{21} + 22q^{22} + 291q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{6} - 9x^{5} - 836x^{4} + 7230x^{3} + 166272x^{2} - 1534153x - 1649494$

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 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
$\operatorname{tr} a_p$	−12	0	9	−5	−11	62	29	215	291	360	664	181	−191	−94	−1726	−345	−12	−336	−402	−1387	811	1645	−1941	−663	1286

$\operatorname{tr} a_n$ 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)[4]   # 134.4.a.e

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

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