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

Newform orbit 134.4.a.b

Trace form

The 2 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$2q + 4q^{2} - 9q^{3} + 8q^{4} - 6q^{5} - 18q^{6} - 21q^{7} + 16q^{8} + 5q^{9} - 12q^{10} - 50q^{11} - 36q^{12} - 81q^{13} - 42q^{14} - 47q^{15} + 32q^{16} - 98q^{17} + 10q^{18} - 75q^{19} - 24q^{20} + 150q^{21} - 100q^{22} - 20q^{23} - 72q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}(\sqrt{37})$ . It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{2} + 15x - 27$

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
$\operatorname{tr} a_p$	4	−9	−6	−21	−50	−81	−98	−75	−20	−26	−240	375	319	489	415	158	−432	−409	−134	−246	−976	−189	−931	−500	270

$\operatorname{tr} a_n$ 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)[1]   # 134.4.a.b

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

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