Atlas › Modular forms › Weight 4 › Level 190 › 190.4.a › 190.4.a.h

Newform orbit 190.4.a.h

Trace form

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

$3q + 6q^{2} - 9q^{3} + 12q^{4} - 15q^{5} - 18q^{6} - 11q^{7} + 24q^{8} - 8q^{9} - 30q^{10} - 68q^{11} - 36q^{12} - 121q^{13} - 22q^{14} + 45q^{15} + 48q^{16} - 103q^{17} - 16q^{18} - 57q^{19} - 60q^{20} - 271q^{21} - 136q^{22} - 203q^{23} - 72q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} + 42x^{2} - 7742x - 405080$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
5	$+1$
19	$+1$

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	−9	−15	−11	−68	−121	−103	−57	−203	91	−154	−304	282	−214	−1144	795	−609	792	−865	−312	79	1806	−34	−1010	−1120

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.17 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, 5, 10, 19, 38, 95, 190]: [1, 2, 3, 7, 6, 17, 32, 94] (certified at ell = 2147483629)
T = [(3, 1), (7, 3)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 5 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(190, 4)[7]   # 190.4.a.h

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

cargo run --release -p sagebrush-web --example atlas -- 190:4 1000