Atlas › Modular forms › Weight 4 › Level 145 › 145.4.a › 145.4.a.b

Newform orbit 145.4.a.b

Trace form

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

$6q - 7q^{2} - 13q^{3} + 17q^{4} + 30q^{5} - 21q^{6} - 79q^{7} + 3q^{8} + 81q^{9} - 35q^{10} - 44q^{11} - 61q^{12} - 53q^{13} + 111q^{14} - 65q^{15} - 75q^{16} - 145q^{17} - 98q^{18} - 84q^{19} + 85q^{20} - 40q^{21} - 310q^{22} - 553q^{23} - 221q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 6. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{6} + 20x^{5} + 67x^{4} - 427x^{3} - 2054x^{2} - 329x + 3896$

Atkin–Lehner signs

$p$	$w_p$
5	$-1$
29	$+1$

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
$\operatorname{tr} a_p$	−7	−13	30	−79	−44	−53	−145	−84	−553	−174	−293	−18	774	−447	−642	−1039	−947	1397	−910	−1184	607	417	−3344	−282	−723

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.116 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, 5, 29, 145]: [1, 3, 9, 48] (certified at ell = 2147483549)
T = [(2, 1), (3, 1)]; h of degree 28 = 1 x 28 (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(145, 4)[1]   # 145.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 145:4 1000