Atlas › Modular forms › Weight 4 › Level 141 › 141.4.a › 141.4.a.b

Newform orbit 141.4.a.b

Trace form

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

$5q - 5q^{2} + 15q^{3} + 7q^{4} - 34q^{5} - 15q^{6} - 26q^{7} - 81q^{8} + 45q^{9} - 32q^{10} - 150q^{11} + 21q^{12} - 50q^{13} - 32q^{14} - 102q^{15} + 243q^{16} - 50q^{17} - 45q^{18} - 234q^{19} - 128q^{20} - 78q^{21} + 136q^{22} - 282q^{23} - 243q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{5}$ , whose minimal polynomial is

$x^{5} + 39x^{4} + 344x^{3} - 3004x^{2} - 53505x - 174563$

Atkin–Lehner signs

$p$	$w_p$
3	$-1$
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 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
$\operatorname{tr} a_p$	−5	15	−34	−26	−150	−50	−50	−234	−282	−354	−262	224	−1124	−366	−235	50	−64	−166	1046	516	−214	3494	−2080	−1066	740

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.117 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, 3, 47, 141]: [1, 2, 13, 50] (certified at ell = 2147483137)
T = [(2, 1), (5, 1)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 4 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(141, 4)[1]   # 141.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 141:4 1000