Atlas › Modular forms › Weight 4 › Level 239 › 239.4.a › 239.4.a.b

Newform orbit 239.4.a.b

Trace form

The 37 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$37q + 4q^{2} + 11q^{3} + 170q^{4} + 43q^{5} + 54q^{6} + 60q^{7} + 27q^{8} + 474q^{9} + 147q^{10} + 55q^{11} + 136q^{12} + 250q^{13} + 169q^{14} + 218q^{15} + 918q^{16} + 189q^{17} + 69q^{18} + 550q^{19} + 486q^{20} + 366q^{21} + 226q^{22} + 74q^{23} + 655q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 37. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{37} - 15x^{36} - 911x^{35} + 13954x^{34} + 361252x^{33} - 5623618x^{32} - 83598787x^{31} + 1308227667x^{30} + 12789770254x^{29} - 197188739887x^{28} - 1388738050889x^{27} + 20435358594063x^{26} + 111901778676198x^{25} - 1503403749091393x^{24} - 6868838597256262x^{23} + 79705297346477658x^{22} + 324328692142421862x^{21} - 3051664537457412835x^{20} - 11714371133621109479x^{19} + 83544342792013661520x^{18} + 317622452426152120757x^{17} - 1596902163126461165915x^{16} - 6272974866942373109781x^{15} + 20393864442433711774337x^{14} + 86528086681031236620165x^{13} - 160383905934810266592252x^{12} - 785785384241772868166524x^{11} + 650314448473378554110644x^{10} + 4305959333510109484347035x^{9} - 682066587236809925042415x^{8} - 12589646313186080405360039x^{7} - 1668160561031770001005467x^{6} + 18357050744681692346284812x^{5} + 2392852785754127948057032x^{4} - 12788079774779435332595217x^{3} + 280844051098231129530930x^{2} + 3125349476742560883195715x - 500496391304187684881350$

Atkin–Lehner signs

$p$	$w_p$
239	$+1$

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
$\operatorname{tr} a_p$	4	11	43	60	55	250	189	550	74	389	1107	1002	1518	1098	1214	904	1658	2313	1535	−3255	3154	2260	−939	1486	6148

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.159 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, 239]: [1, 61] (certified at ell = 2147482093)
T = [(2, 1), (3, 1)]; h of degree 59 = 1 x 59 (phi(ord eps) x dim S_k^new); 8 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(239, 4)[1]   # 239.4.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 239:4 1000