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

Newform orbit 163.4.a.b

Trace form

The 21 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$21q + 12q^{2} + 15q^{3} + 82q^{4} + 79q^{5} + 36q^{6} + 36q^{7} + 144q^{8} + 200q^{9} + 64q^{10} + 71q^{11} + 113q^{12} + 235q^{13} + 39q^{14} + 96q^{15} + 410q^{16} + 428q^{17} + 250q^{18} + 167q^{19} + 595q^{20} + 374q^{21} + 148q^{22} + 276q^{23} + 380q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 21. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{21} - 27x^{20} - 180x^{19} + 10100x^{18} - 18457x^{17} - 1505490x^{16} + 6957783x^{15} + 119283680x^{14} - 722896770x^{13} - 5583954122x^{12} + 38834768042x^{11} + 161569378985x^{10} - 1208721434654x^{9} - 2944664902583x^{8} + 22127542069019x^{7} + 34017967735010x^{6} - 226671144797564x^{5} - 242404116590783x^{4} + 1107994998598080x^{3} + 863420391687518x^{2} - 1465853232899208x - 28790063240196$

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 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
$\operatorname{tr} a_p$	12	15	79	36	71	235	428	167	276	897	596	755	1197	144	79	1884	−257	388	−1135	661	1080	−1638	3194	3322	1191

$\operatorname{tr} a_n$ 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)[1]   # 163.4.a.b

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

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