Atlas › Modular forms › Weight 12 › Level 19 › 19.12.a › 19.12.a.a

Newform orbit 19.12.a.a

Trace form

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

$7q - 9q^{2} + 10q^{3} + 6661q^{4} - 14307q^{5} - 66435q^{6} - 2209q^{7} - 72891q^{8} + 346867q^{9} + 610116q^{10} - 1399461q^{11} - 500945q^{12} - 2639296q^{13} - 6202863q^{14} - 10553670q^{15} - 11910239q^{16} - 9760773q^{17} - 27428622q^{18} + 17332693q^{19} - 60822132q^{20} - 51652462q^{21} - 96770562q^{22} - 117151884q^{23} - 129456201q^{24} + O(q^{25})$

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

$x^{7} - x^{6} - 737511x^{5} - 12159567x^{4} + 131948713182x^{3} - 4449565121016x^{2} - 5287613705621856x + 459570856744281600$

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 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
$\operatorname{tr} a_p$	−9	10	−14307	−2209	−1399461	−2639296	−9760773	17332693	−117151884	−380102178	108083372	470630222	−572622810	1170836039	−3485769735	−9143305584	−847227714	−8144938567	−5797397824	−4538589186	−1815379657	−54907201480	−68609936724	−105124973892	−386940894664

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.506 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]: [2, 20] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 6 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, 12)[0]   # 19.12.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 19:12 1000