Atlas › Modular forms › Weight 4 › Level 118 › 118.4.a › 118.4.a.d

Newform orbit 118.4.a.d

Trace form

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

$3q - 6q^{2} - 12q^{3} + 12q^{4} - 4q^{5} + 24q^{6} + 48q^{7} - 24q^{8} + 33q^{9} + 8q^{10} - 46q^{11} - 48q^{12} - 32q^{13} - 96q^{14} - 77q^{15} + 48q^{16} - 229q^{17} - 66q^{18} - 116q^{19} - 16q^{20} - 243q^{21} + 92q^{22} - 200q^{23} + 96q^{24} + O(q^{25})$

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

$x^{3} + 16x^{2} + 45x - 135$

Atkin–Lehner signs

$p$	$w_p$
2	$+1$
59	$-1$

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	−12	−4	48	−46	−32	−229	−116	−200	−388	36	−166	−88	112	52	−56	177	−274	−1004	−407	−404	2792	282	−214	726

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.113 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, 59, 118]: [1, 2, 16, 47] (certified at ell = 2147482481)
T = [(3, 1), (5, 1)]; h of degree 15 = 1 x 15 (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(118, 4)[3]   # 118.4.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 118:4 1000