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

Newform orbit 197.4.a.a

Trace form

The 22 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$22q - 6q^{2} - 34q^{3} + 68q^{4} - 31q^{5} - 24q^{6} - 102q^{7} - 93q^{8} + 152q^{9} - 133q^{10} - 100q^{11} - 272q^{12} - 223q^{13} - 55q^{14} - 166q^{15} + 112q^{16} - 114q^{17} - 389q^{18} - 529q^{19} - 441q^{20} - 148q^{21} - 671q^{22} - 208q^{23} - 555q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 22. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{22} + 40x^{21} + 329x^{20} - 6072x^{19} - 100431x^{18} + 144531x^{17} + 9083658x^{16} + 19067268x^{15} - 392460045x^{14} - 1438341113x^{13} + 9079515260x^{12} + 42490536383x^{11} - 114097738749x^{10} - 645713381556x^{9} + 735908238004x^{8} + 5231463344257x^{7} - 2185179691603x^{6} - 21701865697784x^{5} + 4032659396506x^{4} + 43218324786288x^{3} - 9787100279456x^{2} - 31803440477952x + 11283007562240$

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 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	−34	−31	−102	−100	−223	−114	−529	−208	−2	−1261	−792	−271	−1349	−986	248	−1281	510	−3070	101	−3352	−1313	−2631	798	−3693

$\operatorname{tr} a_n$ 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)[0]   # 197.4.a.a

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

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