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

Newform orbit 145.4.a.d

Trace form

The 7 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$7q + 6q^{2} + q^{3} + 52q^{4} - 35q^{5} + 3q^{6} + 17q^{7} + 138q^{8} + 118q^{9} - 30q^{10} + 2q^{11} - 77q^{12} - 17q^{13} + 361q^{14} - 5q^{15} + 372q^{16} + 95q^{17} + 261q^{18} + 94q^{19} - 260q^{20} - 100q^{21} + 190q^{22} + 327q^{23} - 117q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 7. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{7} - 7x^{6} - 186x^{5} + 1045x^{4} + 9357x^{3} - 30574x^{2} - 143948x + 154644$

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 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
$\operatorname{tr} a_p$	6	1	−35	17	2	−17	95	94	327	−203	169	−500	1208	−1517	1974	−255	177	−705	−744	2024	−1405	−2157	−158	−2244	−727

$\operatorname{tr} a_n$ 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)[3]   # 145.4.a.d

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

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