Atlas › Modular forms › Weight 4 › Level 163 › 163.4.a › 163.4.a.a

Newform orbit 163.4.a.a

Trace form

The 19 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$19q - 10q^{2} - 15q^{3} + 62q^{4} - 71q^{5} - 36q^{6} - 48q^{7} - 120q^{8} + 110q^{9} - 76q^{10} - 127q^{11} - 247q^{12} - 207q^{13} - 241q^{14} - 144q^{15} + 234q^{16} - 456q^{17} - 344q^{18} - 99q^{19} - 605q^{20} - 466q^{21} - 204q^{22} - 368q^{23} - 484q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 19. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{19} + 25x^{18} - 70x^{17} - 5386x^{16} - 11555x^{15} + 437166x^{14} + 1329443x^{13} - 17500984x^{12} - 47366076x^{11} + 375189648x^{10} + 667335034x^{9} - 4189738957x^{8} - 2766769810x^{7} + 20728835229x^{6} - 6487780125x^{5} - 30289921070x^{4} + 24707146682x^{3} - 178333913x^{2} - 2254979968x - 12056458$

Atkin–Lehner signs

$p$	$w_p$
163	$-1$

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
$\operatorname{tr} a_p$	−10	−15	−71	−48	−127	−207	−456	−99	−368	−669	−24	−1095	−1305	−374	−1647	−1790	149	−1294	463	−273	−420	3018	−3408	−1342	−3483

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.12 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, 163]: [1, 42] (certified at ell = 2147483179)
T = [(2, 1), (3, 1)]; h of degree 40 = 1 x 40 (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(163, 4)[0]   # 163.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 163:4 1000