Atlas › Modular forms › Weight 4 › Level 197 › 197.4.a › 197.4.a.b

Newform orbit 197.4.a.b

Trace form

The 27 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$27q + 4q^{2} + 32q^{3} + 128q^{4} + 29q^{5} + 36q^{6} + 122q^{7} + 27q^{8} + 287q^{9} + 127q^{10} + 98q^{11} + 256q^{12} + 193q^{13} + 113q^{14} + 194q^{15} + 672q^{16} + 124q^{17} + 61q^{18} + 535q^{19} + 279q^{20} + 188q^{21} + 165q^{22} + 344q^{23} + 165q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 27. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{27} - 36x^{26} - 68x^{25} + 16252x^{24} - 112139x^{23} - 2851397x^{22} + 32647270x^{21} + 244213597x^{20} - 4189266744x^{19} - 9464930368x^{18} + 307196007987x^{17} - 23617252088x^{16} - 13999868912662x^{15} + 17525045280228x^{14} + 409380157020981x^{13} - 723530049661428x^{12} - 7743074206694232x^{11} + 13751046737341387x^{10} + 94245991494291925x^{9} - 124410009485050507x^{8} - 719251601102723307x^{7} + 372588494689392154x^{6} + 3048359668868492862x^{5} + 1011598092283840628x^{4} - 4593769899760536280x^{3} - 3914396346055392752x^{2} + 315610957355591072x + 768305593258318400$

Atkin–Lehner signs

$p$	$w_p$
197	$+1$

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
$\operatorname{tr} a_p$	4	32	29	122	98	193	124	535	344	346	599	1428	549	2091	1270	−208	603	−314	4372	−871	942	−197	−799	−572	6335

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.139 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, 197]: [1, 51] (certified at ell = 2147479489)
T = [(2, 1), (3, 1)]; h of degree 49 = 1 x 49 (phi(ord eps) x dim S_k^new); 7 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(197, 4)[1]   # 197.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 197:4 1000