Atlas › Modular forms › Weight 2 › Level 197 › 197.2.a › 197.2.a.c

Newform orbit 197.2.a.c

Trace form

The 10 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$10q + 10q^{3} + 10q^{4} + 2q^{5} - 4q^{6} + 11q^{7} - 3q^{8} + 12q^{9} - 2q^{10} + 2q^{11} + 4q^{12} + 8q^{13} - 9q^{14} - q^{15} - 2q^{16} - 3q^{17} - 9q^{18} + 17q^{19} - 2q^{20} - 2q^{21} + 3q^{22} - 4q^{23} - 15q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 10. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{10} - 10x^{9} + 18x^{8} + 113x^{7} - 463x^{6} + 182x^{5} + 1642x^{4} - 3208x^{3} + 2475x^{2} - 863x + 112$

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
$\operatorname{tr} a_p$	0	10	2	11	2	8	−3	17	−4	−9	20	12	−18	11	5	−6	−1	4	46	−5	25	−1	28	−6	2

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.247 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 197]: [0, 17] (certified at ell = 2147479489)
T = [(2, 1), (3, 1)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 2 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, 2)[2]   # 197.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 197:2 1000