Atlas › Modular forms › Weight 4 › Level 193 › 193.4.a › 193.4.a.a

Newform orbit 193.4.a.a

Trace form

The 23 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$23q - 10q^{2} - 26q^{3} + 78q^{4} - 60q^{5} - 34q^{6} - 82q^{7} - 120q^{8} + 185q^{9} - 66q^{10} - 180q^{11} - 339q^{12} - 128q^{13} - 165q^{14} - 254q^{15} + 326q^{16} - 344q^{17} - 410q^{18} - 194q^{19} - 892q^{20} - 148q^{21} - 275q^{22} - 926q^{23} - 276q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 23. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{23} + 36x^{22} + 148x^{21} - 8133x^{20} - 82535x^{19} + 608446x^{18} + 9655058x^{17} - 17048295x^{16} - 540193314x^{15} - 81092688x^{14} + 16913944389x^{13} + 16832883068x^{12} - 305492922700x^{11} - 449962571305x^{10} + 3027752565671x^{9} + 5921806891260x^{8} - 13628691327810x^{7} - 39915267687345x^{6} + 4001608637093x^{5} + 100838506017392x^{4} + 118113462130683x^{3} + 50711989117005x^{2} + 5641033152000x - 248677774125$

Atkin–Lehner signs

$p$	$w_p$
193	$-1$

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	−26	−60	−82	−180	−128	−344	−194	−926	−254	−770	−664	−1120	−932	−2950	−906	−360	−914	460	−2466	872	−358	−3752	−3058	72

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.138 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, 193]: [1, 50] (certified at ell = 2147483137)
T = [(2, 1), (3, 1)]; h of degree 48 = 1 x 48 (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(193, 4)[0]   # 193.4.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 193:4 1000