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

Newform orbit 193.4.a.b

Trace form

The 25 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$25q + 10q^{2} + 22q^{3} + 102q^{4} + 50q^{5} + 38q^{6} + 86q^{7} + 120q^{8} + 275q^{9} + 34q^{10} + 128q^{11} + 141q^{12} + 54q^{13} + 171q^{14} + 286q^{15} + 550q^{16} + 336q^{17} + 130q^{18} + 186q^{19} + 428q^{20} + 188q^{21} + 121q^{22} + 822q^{23} + 588q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 25. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{25} - 32x^{24} - 152x^{23} + 14307x^{22} - 61687x^{21} - 2543390x^{20} + 20512270x^{19} + 232597141x^{18} - 2605586242x^{17} - 11660950688x^{16} + 182086506497x^{15} + 303030190564x^{14} - 7730590822100x^{13} - 2648375596321x^{12} + 206307070504203x^{11} - 42008689848676x^{10} - 3457120130777422x^{9} + 822045288476995x^{8} + 34951127230743153x^{7} + 3412401749523912x^{6} - 188924899517238917x^{5} - 118757972954840027x^{4} + 372306833758587672x^{3} + 371847600483054299x^{2} - 75288552522086368x - 103745154310658704$

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 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
$\operatorname{tr} a_p$	10	22	50	86	128	54	336	186	822	384	594	446	356	444	3536	344	1272	−668	−1900	1768	−664	84	4008	110	2440

$\operatorname{tr} a_n$ 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)[1]   # 193.4.a.b

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

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