from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2243, base_ring=CyclotomicField(2242))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,2243))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2243\) | |
| Conductor: | \(2243\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2242\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{1121})$ |
| Fixed field: | Number field defined by a degree 2242 polynomial (not computed) |
First 31 of 1044 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2243}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{2242}\right)\) | \(e\left(\frac{73}{1121}\right)\) | \(e\left(\frac{1}{1121}\right)\) | \(e\left(\frac{1363}{2242}\right)\) | \(e\left(\frac{147}{2242}\right)\) | \(e\left(\frac{1070}{1121}\right)\) | \(e\left(\frac{3}{2242}\right)\) | \(e\left(\frac{146}{1121}\right)\) | \(e\left(\frac{682}{1121}\right)\) | \(e\left(\frac{1014}{1121}\right)\) |
| \(\chi_{2243}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1363}{2242}\right)\) | \(e\left(\frac{851}{1121}\right)\) | \(e\left(\frac{242}{1121}\right)\) | \(e\left(\frac{1393}{2242}\right)\) | \(e\left(\frac{823}{2242}\right)\) | \(e\left(\frac{1110}{1121}\right)\) | \(e\left(\frac{1847}{2242}\right)\) | \(e\left(\frac{581}{1121}\right)\) | \(e\left(\frac{257}{1121}\right)\) | \(e\left(\frac{1010}{1121}\right)\) |
| \(\chi_{2243}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{147}{2242}\right)\) | \(e\left(\frac{642}{1121}\right)\) | \(e\left(\frac{147}{1121}\right)\) | \(e\left(\frac{823}{2242}\right)\) | \(e\left(\frac{1431}{2242}\right)\) | \(e\left(\frac{350}{1121}\right)\) | \(e\left(\frac{441}{2242}\right)\) | \(e\left(\frac{163}{1121}\right)\) | \(e\left(\frac{485}{1121}\right)\) | \(e\left(\frac{1086}{1121}\right)\) |
| \(\chi_{2243}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{2242}\right)\) | \(e\left(\frac{219}{1121}\right)\) | \(e\left(\frac{3}{1121}\right)\) | \(e\left(\frac{1847}{2242}\right)\) | \(e\left(\frac{441}{2242}\right)\) | \(e\left(\frac{968}{1121}\right)\) | \(e\left(\frac{9}{2242}\right)\) | \(e\left(\frac{438}{1121}\right)\) | \(e\left(\frac{925}{1121}\right)\) | \(e\left(\frac{800}{1121}\right)\) |
| \(\chi_{2243}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1865}{2242}\right)\) | \(e\left(\frac{504}{1121}\right)\) | \(e\left(\frac{744}{1121}\right)\) | \(e\left(\frac{1809}{2242}\right)\) | \(e\left(\frac{631}{2242}\right)\) | \(e\left(\frac{170}{1121}\right)\) | \(e\left(\frac{1111}{2242}\right)\) | \(e\left(\frac{1008}{1121}\right)\) | \(e\left(\frac{716}{1121}\right)\) | \(e\left(\frac{1104}{1121}\right)\) |
| \(\chi_{2243}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2141}{2242}\right)\) | \(e\left(\frac{474}{1121}\right)\) | \(e\left(\frac{1020}{1121}\right)\) | \(e\left(\frac{1341}{2242}\right)\) | \(e\left(\frac{847}{2242}\right)\) | \(e\left(\frac{667}{1121}\right)\) | \(e\left(\frac{1939}{2242}\right)\) | \(e\left(\frac{948}{1121}\right)\) | \(e\left(\frac{620}{1121}\right)\) | \(e\left(\frac{718}{1121}\right)\) |
| \(\chi_{2243}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1509}{2242}\right)\) | \(e\left(\frac{299}{1121}\right)\) | \(e\left(\frac{388}{1121}\right)\) | \(e\left(\frac{853}{2242}\right)\) | \(e\left(\frac{2107}{2242}\right)\) | \(e\left(\frac{390}{1121}\right)\) | \(e\left(\frac{43}{2242}\right)\) | \(e\left(\frac{598}{1121}\right)\) | \(e\left(\frac{60}{1121}\right)\) | \(e\left(\frac{1082}{1121}\right)\) |
| \(\chi_{2243}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{293}{2242}\right)\) | \(e\left(\frac{90}{1121}\right)\) | \(e\left(\frac{293}{1121}\right)\) | \(e\left(\frac{283}{2242}\right)\) | \(e\left(\frac{473}{2242}\right)\) | \(e\left(\frac{751}{1121}\right)\) | \(e\left(\frac{879}{2242}\right)\) | \(e\left(\frac{180}{1121}\right)\) | \(e\left(\frac{288}{1121}\right)\) | \(e\left(\frac{37}{1121}\right)\) |
| \(\chi_{2243}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{329}{2242}\right)\) | \(e\left(\frac{476}{1121}\right)\) | \(e\left(\frac{329}{1121}\right)\) | \(e\left(\frac{27}{2242}\right)\) | \(e\left(\frac{1281}{2242}\right)\) | \(e\left(\frac{36}{1121}\right)\) | \(e\left(\frac{987}{2242}\right)\) | \(e\left(\frac{952}{1121}\right)\) | \(e\left(\frac{178}{1121}\right)\) | \(e\left(\frac{669}{1121}\right)\) |
| \(\chi_{2243}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1365}{2242}\right)\) | \(e\left(\frac{997}{1121}\right)\) | \(e\left(\frac{244}{1121}\right)\) | \(e\left(\frac{1877}{2242}\right)\) | \(e\left(\frac{1117}{2242}\right)\) | \(e\left(\frac{1008}{1121}\right)\) | \(e\left(\frac{1853}{2242}\right)\) | \(e\left(\frac{873}{1121}\right)\) | \(e\left(\frac{500}{1121}\right)\) | \(e\left(\frac{796}{1121}\right)\) |
| \(\chi_{2243}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2029}{2242}\right)\) | \(e\left(\frac{145}{1121}\right)\) | \(e\left(\frac{908}{1121}\right)\) | \(e\left(\frac{1141}{2242}\right)\) | \(e\left(\frac{77}{2242}\right)\) | \(e\left(\frac{774}{1121}\right)\) | \(e\left(\frac{1603}{2242}\right)\) | \(e\left(\frac{290}{1121}\right)\) | \(e\left(\frac{464}{1121}\right)\) | \(e\left(\frac{371}{1121}\right)\) |
| \(\chi_{2243}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2231}{2242}\right)\) | \(e\left(\frac{318}{1121}\right)\) | \(e\left(\frac{1110}{1121}\right)\) | \(e\left(\frac{701}{2242}\right)\) | \(e\left(\frac{625}{2242}\right)\) | \(e\left(\frac{561}{1121}\right)\) | \(e\left(\frac{2209}{2242}\right)\) | \(e\left(\frac{636}{1121}\right)\) | \(e\left(\frac{345}{1121}\right)\) | \(e\left(\frac{56}{1121}\right)\) |
| \(\chi_{2243}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{2242}\right)\) | \(e\left(\frac{788}{1121}\right)\) | \(e\left(\frac{149}{1121}\right)\) | \(e\left(\frac{1307}{2242}\right)\) | \(e\left(\frac{1725}{2242}\right)\) | \(e\left(\frac{248}{1121}\right)\) | \(e\left(\frac{447}{2242}\right)\) | \(e\left(\frac{455}{1121}\right)\) | \(e\left(\frac{728}{1121}\right)\) | \(e\left(\frac{872}{1121}\right)\) |
| \(\chi_{2243}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{559}{2242}\right)\) | \(e\left(\frac{451}{1121}\right)\) | \(e\left(\frac{559}{1121}\right)\) | \(e\left(\frac{1879}{2242}\right)\) | \(e\left(\frac{1461}{2242}\right)\) | \(e\left(\frac{637}{1121}\right)\) | \(e\left(\frac{1677}{2242}\right)\) | \(e\left(\frac{902}{1121}\right)\) | \(e\left(\frac{98}{1121}\right)\) | \(e\left(\frac{721}{1121}\right)\) |
| \(\chi_{2243}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{2242}\right)\) | \(e\left(\frac{365}{1121}\right)\) | \(e\left(\frac{5}{1121}\right)\) | \(e\left(\frac{89}{2242}\right)\) | \(e\left(\frac{735}{2242}\right)\) | \(e\left(\frac{866}{1121}\right)\) | \(e\left(\frac{15}{2242}\right)\) | \(e\left(\frac{730}{1121}\right)\) | \(e\left(\frac{47}{1121}\right)\) | \(e\left(\frac{586}{1121}\right)\) |
| \(\chi_{2243}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{2242}\right)\) | \(e\left(\frac{407}{1121}\right)\) | \(e\left(\frac{67}{1121}\right)\) | \(e\left(\frac{1641}{2242}\right)\) | \(e\left(\frac{881}{2242}\right)\) | \(e\left(\frac{1067}{1121}\right)\) | \(e\left(\frac{201}{2242}\right)\) | \(e\left(\frac{814}{1121}\right)\) | \(e\left(\frac{854}{1121}\right)\) | \(e\left(\frac{678}{1121}\right)\) |
| \(\chi_{2243}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1261}{2242}\right)\) | \(e\left(\frac{131}{1121}\right)\) | \(e\left(\frac{140}{1121}\right)\) | \(e\left(\frac{1371}{2242}\right)\) | \(e\left(\frac{1523}{2242}\right)\) | \(e\left(\frac{707}{1121}\right)\) | \(e\left(\frac{1541}{2242}\right)\) | \(e\left(\frac{262}{1121}\right)\) | \(e\left(\frac{195}{1121}\right)\) | \(e\left(\frac{714}{1121}\right)\) |
| \(\chi_{2243}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1841}{2242}\right)\) | \(e\left(\frac{994}{1121}\right)\) | \(e\left(\frac{720}{1121}\right)\) | \(e\left(\frac{485}{2242}\right)\) | \(e\left(\frac{1587}{2242}\right)\) | \(e\left(\frac{273}{1121}\right)\) | \(e\left(\frac{1039}{2242}\right)\) | \(e\left(\frac{867}{1121}\right)\) | \(e\left(\frac{42}{1121}\right)\) | \(e\left(\frac{309}{1121}\right)\) |
| \(\chi_{2243}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2011}{2242}\right)\) | \(e\left(\frac{1073}{1121}\right)\) | \(e\left(\frac{890}{1121}\right)\) | \(e\left(\frac{1269}{2242}\right)\) | \(e\left(\frac{1915}{2242}\right)\) | \(e\left(\frac{571}{1121}\right)\) | \(e\left(\frac{1549}{2242}\right)\) | \(e\left(\frac{1025}{1121}\right)\) | \(e\left(\frac{519}{1121}\right)\) | \(e\left(\frac{55}{1121}\right)\) |
| \(\chi_{2243}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1241}{2242}\right)\) | \(e\left(\frac{913}{1121}\right)\) | \(e\left(\frac{120}{1121}\right)\) | \(e\left(\frac{1015}{2242}\right)\) | \(e\left(\frac{825}{2242}\right)\) | \(e\left(\frac{606}{1121}\right)\) | \(e\left(\frac{1481}{2242}\right)\) | \(e\left(\frac{705}{1121}\right)\) | \(e\left(\frac{7}{1121}\right)\) | \(e\left(\frac{612}{1121}\right)\) |
| \(\chi_{2243}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{2242}\right)\) | \(e\left(\frac{1043}{1121}\right)\) | \(e\left(\frac{45}{1121}\right)\) | \(e\left(\frac{801}{2242}\right)\) | \(e\left(\frac{2131}{2242}\right)\) | \(e\left(\frac{1068}{1121}\right)\) | \(e\left(\frac{135}{2242}\right)\) | \(e\left(\frac{965}{1121}\right)\) | \(e\left(\frac{423}{1121}\right)\) | \(e\left(\frac{790}{1121}\right)\) |
| \(\chi_{2243}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1655}{2242}\right)\) | \(e\left(\frac{868}{1121}\right)\) | \(e\left(\frac{534}{1121}\right)\) | \(e\left(\frac{313}{2242}\right)\) | \(e\left(\frac{1149}{2242}\right)\) | \(e\left(\frac{791}{1121}\right)\) | \(e\left(\frac{481}{2242}\right)\) | \(e\left(\frac{615}{1121}\right)\) | \(e\left(\frac{984}{1121}\right)\) | \(e\left(\frac{33}{1121}\right)\) |
| \(\chi_{2243}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1715}{2242}\right)\) | \(e\left(\frac{764}{1121}\right)\) | \(e\left(\frac{594}{1121}\right)\) | \(e\left(\frac{1381}{2242}\right)\) | \(e\left(\frac{1001}{2242}\right)\) | \(e\left(\frac{1094}{1121}\right)\) | \(e\left(\frac{661}{2242}\right)\) | \(e\left(\frac{407}{1121}\right)\) | \(e\left(\frac{427}{1121}\right)\) | \(e\left(\frac{339}{1121}\right)\) |
| \(\chi_{2243}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{485}{2242}\right)\) | \(e\left(\frac{654}{1121}\right)\) | \(e\left(\frac{485}{1121}\right)\) | \(e\left(\frac{1907}{2242}\right)\) | \(e\left(\frac{1793}{2242}\right)\) | \(e\left(\frac{1048}{1121}\right)\) | \(e\left(\frac{1455}{2242}\right)\) | \(e\left(\frac{187}{1121}\right)\) | \(e\left(\frac{75}{1121}\right)\) | \(e\left(\frac{792}{1121}\right)\) |
| \(\chi_{2243}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1867}{2242}\right)\) | \(e\left(\frac{650}{1121}\right)\) | \(e\left(\frac{746}{1121}\right)\) | \(e\left(\frac{51}{2242}\right)\) | \(e\left(\frac{925}{2242}\right)\) | \(e\left(\frac{68}{1121}\right)\) | \(e\left(\frac{1117}{2242}\right)\) | \(e\left(\frac{179}{1121}\right)\) | \(e\left(\frac{959}{1121}\right)\) | \(e\left(\frac{890}{1121}\right)\) |
| \(\chi_{2243}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{439}{2242}\right)\) | \(e\left(\frac{659}{1121}\right)\) | \(e\left(\frac{439}{1121}\right)\) | \(e\left(\frac{1985}{2242}\right)\) | \(e\left(\frac{1757}{2242}\right)\) | \(e\left(\frac{31}{1121}\right)\) | \(e\left(\frac{1317}{2242}\right)\) | \(e\left(\frac{197}{1121}\right)\) | \(e\left(\frac{91}{1121}\right)\) | \(e\left(\frac{109}{1121}\right)\) |
| \(\chi_{2243}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1149}{2242}\right)\) | \(e\left(\frac{923}{1121}\right)\) | \(e\left(\frac{28}{1121}\right)\) | \(e\left(\frac{1171}{2242}\right)\) | \(e\left(\frac{753}{2242}\right)\) | \(e\left(\frac{814}{1121}\right)\) | \(e\left(\frac{1205}{2242}\right)\) | \(e\left(\frac{725}{1121}\right)\) | \(e\left(\frac{39}{1121}\right)\) | \(e\left(\frac{367}{1121}\right)\) |
| \(\chi_{2243}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2143}{2242}\right)\) | \(e\left(\frac{620}{1121}\right)\) | \(e\left(\frac{1022}{1121}\right)\) | \(e\left(\frac{1825}{2242}\right)\) | \(e\left(\frac{1141}{2242}\right)\) | \(e\left(\frac{565}{1121}\right)\) | \(e\left(\frac{1945}{2242}\right)\) | \(e\left(\frac{119}{1121}\right)\) | \(e\left(\frac{863}{1121}\right)\) | \(e\left(\frac{504}{1121}\right)\) |
| \(\chi_{2243}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{791}{2242}\right)\) | \(e\left(\frac{572}{1121}\right)\) | \(e\left(\frac{791}{1121}\right)\) | \(e\left(\frac{1973}{2242}\right)\) | \(e\left(\frac{1935}{2242}\right)\) | \(e\left(\frac{15}{1121}\right)\) | \(e\left(\frac{131}{2242}\right)\) | \(e\left(\frac{23}{1121}\right)\) | \(e\left(\frac{261}{1121}\right)\) | \(e\left(\frac{559}{1121}\right)\) |
| \(\chi_{2243}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1511}{2242}\right)\) | \(e\left(\frac{445}{1121}\right)\) | \(e\left(\frac{390}{1121}\right)\) | \(e\left(\frac{1337}{2242}\right)\) | \(e\left(\frac{159}{2242}\right)\) | \(e\left(\frac{288}{1121}\right)\) | \(e\left(\frac{49}{2242}\right)\) | \(e\left(\frac{890}{1121}\right)\) | \(e\left(\frac{303}{1121}\right)\) | \(e\left(\frac{868}{1121}\right)\) |
| \(\chi_{2243}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{613}{2242}\right)\) | \(e\left(\frac{1030}{1121}\right)\) | \(e\left(\frac{613}{1121}\right)\) | \(e\left(\frac{1495}{2242}\right)\) | \(e\left(\frac{431}{2242}\right)\) | \(e\left(\frac{125}{1121}\right)\) | \(e\left(\frac{1839}{2242}\right)\) | \(e\left(\frac{939}{1121}\right)\) | \(e\left(\frac{1054}{1121}\right)\) | \(e\left(\frac{548}{1121}\right)\) |