Properties

Label 4.4.19429.1-21.1-g
Base field 4.4.19429.1
Weight $[2, 2, 2, 2]$
Level norm $21$
Level $[21, 21, w^{2} - w - 6]$
Dimension $9$
CM no
Base change no

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Base field 4.4.19429.1

Generator \(w\), with minimal polynomial \(x^{4} - x^{3} - 7x^{2} - x + 5\); narrow class number \(1\) and class number \(1\).

Form

Weight: $[2, 2, 2, 2]$
Level: $[21, 21, w^{2} - w - 6]$
Dimension: $9$
CM: no
Base change: no
Newspace dimension: $37$

Hecke eigenvalues ($q$-expansion)

The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:

\(x^{9} - 2x^{8} - 18x^{7} + 36x^{6} + 85x^{5} - 182x^{4} - 81x^{3} + 225x^{2} - 16x - 24\)

  Show full eigenvalues   Hide large eigenvalues

Norm Prime Eigenvalue
3 $[3, 3, -w^{3} + 2w^{2} + 5w - 3]$ $\phantom{-}1$
5 $[5, 5, w]$ $\phantom{-}e$
7 $[7, 7, -w^{3} + 2w^{2} + 4w - 3]$ $\phantom{-}1$
13 $[13, 13, -w^{2} + w + 4]$ $\phantom{-}\frac{229}{1687}e^{8} - \frac{121}{1687}e^{7} - \frac{4359}{1687}e^{6} + \frac{2146}{1687}e^{5} + \frac{23264}{1687}e^{4} - \frac{12577}{1687}e^{3} - \frac{5215}{241}e^{2} + \frac{17974}{1687}e + \frac{9917}{1687}$
13 $[13, 13, -w^{3} + 2w^{2} + 5w - 2]$ $\phantom{-}\frac{167}{1687}e^{8} + \frac{258}{1687}e^{7} - \frac{3142}{1687}e^{6} - \frac{4520}{1687}e^{5} + \frac{16133}{1687}e^{4} + \frac{18159}{1687}e^{3} - \frac{3681}{241}e^{2} - \frac{17774}{1687}e + \frac{6488}{1687}$
16 $[16, 2, 2]$ $\phantom{-}\frac{841}{3374}e^{8} - \frac{108}{1687}e^{7} - \frac{7669}{1687}e^{6} + \frac{1539}{1687}e^{5} + \frac{74335}{3374}e^{4} - \frac{8033}{1687}e^{3} - \frac{12785}{482}e^{2} + \frac{953}{3374}e + \frac{305}{1687}$
17 $[17, 17, -w + 2]$ $-\frac{276}{1687}e^{8} + \frac{109}{1687}e^{7} + \frac{4819}{1687}e^{6} - \frac{1975}{1687}e^{5} - \frac{20915}{1687}e^{4} + \frac{13184}{1687}e^{3} + \frac{2483}{241}e^{2} - \frac{17265}{1687}e + \frac{6612}{1687}$
19 $[19, 19, -w^{3} + 2w^{2} + 3w - 2]$ $\phantom{-}\frac{228}{1687}e^{8} + \frac{130}{1687}e^{7} - \frac{4421}{1687}e^{6} - \frac{2696}{1687}e^{5} + \frac{24319}{1687}e^{4} + \frac{13387}{1687}e^{3} - \frac{5750}{241}e^{2} - \frac{19771}{1687}e + \frac{8474}{1687}$
27 $[27, 3, w^{3} - 3w^{2} - 4w + 7]$ $-\frac{61}{1687}e^{8} + \frac{128}{1687}e^{7} + \frac{1279}{1687}e^{6} - \frac{1824}{1687}e^{5} - \frac{8186}{1687}e^{4} + \frac{4772}{1687}e^{3} + \frac{2310}{241}e^{2} + \frac{1997}{1687}e - \frac{5360}{1687}$
31 $[31, 31, w^{3} - 3w^{2} - 3w + 7]$ $-\frac{41}{1687}e^{8} + \frac{169}{1687}e^{7} + \frac{832}{1687}e^{6} - \frac{2830}{1687}e^{5} - \frac{5668}{1687}e^{4} + \frac{11836}{1687}e^{3} + \frac{2165}{241}e^{2} - \frac{9001}{1687}e - \frac{10240}{1687}$
31 $[31, 31, -w^{3} + w^{2} + 6w + 1]$ $-\frac{291}{1687}e^{8} + \frac{500}{1687}e^{7} + \frac{5576}{1687}e^{6} - \frac{8812}{1687}e^{5} - \frac{30395}{1687}e^{4} + \frac{43313}{1687}e^{3} + \frac{6508}{241}e^{2} - \frac{50348}{1687}e - \frac{6598}{1687}$
41 $[41, 41, w^{2} - w - 1]$ $-\frac{150}{241}e^{8} + \frac{54}{241}e^{7} + \frac{2750}{241}e^{6} - \frac{890}{241}e^{5} - \frac{13583}{241}e^{4} + \frac{5101}{241}e^{3} + \frac{18337}{241}e^{2} - \frac{4757}{241}e - \frac{3888}{241}$
43 $[43, 43, 2w^{3} - 5w^{2} - 6w + 4]$ $-\frac{228}{1687}e^{8} - \frac{130}{1687}e^{7} + \frac{4421}{1687}e^{6} + \frac{2696}{1687}e^{5} - \frac{24319}{1687}e^{4} - \frac{13387}{1687}e^{3} + \frac{5509}{241}e^{2} + \frac{18084}{1687}e + \frac{3335}{1687}$
47 $[47, 47, -w^{3} + 2w^{2} + 5w - 1]$ $\phantom{-}\frac{229}{1687}e^{8} - \frac{121}{1687}e^{7} - \frac{4359}{1687}e^{6} + \frac{2146}{1687}e^{5} + \frac{23264}{1687}e^{4} - \frac{12577}{1687}e^{3} - \frac{4974}{241}e^{2} + \frac{17974}{1687}e + \frac{6543}{1687}$
53 $[53, 53, -w - 3]$ $-\frac{1301}{3374}e^{8} + \frac{480}{1687}e^{7} + \frac{11966}{1687}e^{6} - \frac{8527}{1687}e^{5} - \frac{118753}{3374}e^{4} + \frac{46574}{1687}e^{3} + \frac{21663}{482}e^{2} - \frac{105643}{3374}e - \frac{4917}{1687}$
53 $[53, 53, -w^{3} + 3w^{2} + 3w - 6]$ $\phantom{-}\frac{251}{482}e^{8} - \frac{50}{241}e^{7} - \frac{2341}{241}e^{6} + \frac{833}{241}e^{5} + \frac{23913}{482}e^{4} - \frac{4500}{241}e^{3} - \frac{33645}{482}e^{2} + \frac{7033}{482}e + \frac{2877}{241}$
59 $[59, 59, 2w^{2} - 3w - 6]$ $-\frac{625}{3374}e^{8} - \frac{8}{1687}e^{7} + \frac{5930}{1687}e^{6} + \frac{114}{1687}e^{5} - \frac{62661}{3374}e^{4} + \frac{2654}{1687}e^{3} + \frac{13629}{482}e^{2} - \frac{6365}{3374}e - \frac{3039}{1687}$
59 $[59, 59, w^{3} - w^{2} - 7w - 3]$ $-\frac{337}{1687}e^{8} + \frac{237}{1687}e^{7} + \frac{6098}{1687}e^{6} - \frac{3799}{1687}e^{5} - \frac{29101}{1687}e^{4} + \frac{17956}{1687}e^{3} + \frac{4793}{241}e^{2} - \frac{20329}{1687}e - \frac{435}{1687}$
79 $[79, 79, 2w^{3} - 4w^{2} - 8w + 3]$ $-\frac{333}{3374}e^{8} + \frac{460}{1687}e^{7} + \frac{3173}{1687}e^{6} - \frac{8242}{1687}e^{5} - \frac{33321}{3374}e^{4} + \frac{39713}{1687}e^{3} + \frac{5005}{482}e^{2} - \frac{92033}{3374}e + \frac{8573}{1687}$
79 $[79, 79, w^{2} - 2w - 1]$ $\phantom{-}\frac{1629}{3374}e^{8} - \frac{1156}{1687}e^{7} - \frac{15294}{1687}e^{6} + \frac{19847}{1687}e^{5} + \frac{157349}{3374}e^{4} - \frac{92231}{1687}e^{3} - \frac{27897}{482}e^{2} + \frac{170903}{3374}e + \frac{2015}{1687}$
Display number of eigenvalues

Atkin-Lehner eigenvalues

Norm Prime Eigenvalue
$3$ $[3, 3, -w^{3} + 2w^{2} + 5w - 3]$ $-1$
$7$ $[7, 7, -w^{3} + 2w^{2} + 4w - 3]$ $-1$