Atlas › Modular forms › Weight 4 › Level 173 › 173.4.a › 173.4.a.b

Newform orbit 173.4.a.b

Trace form

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

$25q + 3q^{2} + 26q^{3} + 133q^{4} + 15q^{5} + 24q^{6} + 103q^{7} + 57q^{8} + 303q^{9} + 54q^{10} + 148q^{11} + 196q^{12} + 128q^{13} + 149q^{14} + 176q^{15} + 737q^{16} + 150q^{17} + 24q^{18} + 298q^{19} + 8q^{20} + 144q^{21} + 523q^{22} + 528q^{23} + 205q^{24} + O(q^{25})$

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

$x^{25} - 29x^{24} - 259x^{23} + 14290x^{22} - 18751x^{21} - 2925736x^{20} + 14540710x^{19} + 326326088x^{18} - 2335503283x^{17} - 21810686966x^{16} + 196485848316x^{15} + 901290971370x^{14} - 10014849795776x^{13} - 22768372601518x^{12} + 322936873654407x^{11} + 331987362294921x^{10} - 6590537546526749x^{9} - 2490145600089452x^{8} + 82333206073222035x^{7} + 10335659358163956x^{6} - 581272065486111806x^{5} - 96425316242932604x^{4} + 1907207944319673531x^{3} + 749441085717784104x^{2} - 1292569389284967084x - 665869250741185568$

Atkin–Lehner signs

$p$	$w_p$
173	$+1$

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
$\operatorname{tr} a_p$	3	26	15	103	148	128	150	298	528	50	258	1356	305	2485	248	423	−1990	647	3273	−555	423	1365	531	−81	3076

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.125 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, 173]: [1, 45] (certified at ell = 2147482093)
T = [(2, 1), (3, 1)]; h of degree 43 = 1 x 43 (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(173, 4)[1]   # 173.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 173:4 1000