Atlas › Modular forms › Weight 4 › Level 19 › 19.4.a › 19.4.a.b

Newform orbit 19.4.a.b

Trace form

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

$3q + 3q^{2} + q^{3} + 21q^{4} + 14q^{5} - 65q^{6} - 35q^{7} + 27q^{8} + 48q^{9} - 88q^{10} + 16q^{11} - 115q^{12} + 65q^{13} + 37q^{14} + 140q^{15} + 33q^{16} + 29q^{17} + 138q^{18} - 57q^{19} + 100q^{20} - 25q^{21} + 118q^{22} - 101q^{23} - 489q^{24} + O(q^{25})$

Coefficient field

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

$x^{3} - 4x^{2} - 14x + 48$

Atkin–Lehner signs

$p$	$w_p$
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 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
$\operatorname{tr} a_p$	3	1	14	−35	16	65	29	−57	−101	377	−140	−290	956	−570	66	817	265	988	−207	846	627	382	−766	−172	−2450

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.091 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, 19]: [1, 6] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 4 = 1 x 4 (phi(ord eps) x dim S_k^new); 1 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(19, 4)[1]   # 19.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 19:4 1000