from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3712, base_ring=CyclotomicField(224))
M = H._module
chi = DirichletCharacter(H, M([112,147,200]))
chi.galois_orbit()
[g,chi] = znchar(Mod(11,3712))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3712\) | |
| Conductor: | \(3712\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(224\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{224})$ |
| Fixed field: | Number field defined by a degree 224 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3712}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{224}\right)\) | \(e\left(\frac{67}{224}\right)\) | \(e\left(\frac{87}{112}\right)\) | \(e\left(\frac{97}{112}\right)\) | \(e\left(\frac{135}{224}\right)\) | \(e\left(\frac{205}{224}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{141}{224}\right)\) | \(e\left(\frac{159}{224}\right)\) |
| \(\chi_{3712}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{224}\right)\) | \(e\left(\frac{27}{224}\right)\) | \(e\left(\frac{15}{112}\right)\) | \(e\left(\frac{9}{112}\right)\) | \(e\left(\frac{31}{224}\right)\) | \(e\left(\frac{213}{224}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{117}{224}\right)\) | \(e\left(\frac{151}{224}\right)\) |
| \(\chi_{3712}(131,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{135}{224}\right)\) | \(e\left(\frac{69}{224}\right)\) | \(e\left(\frac{1}{112}\right)\) | \(e\left(\frac{23}{112}\right)\) | \(e\left(\frac{129}{224}\right)\) | \(e\left(\frac{171}{224}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{75}{224}\right)\) | \(e\left(\frac{137}{224}\right)\) |
| \(\chi_{3712}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{224}\right)\) | \(e\left(\frac{221}{224}\right)\) | \(e\left(\frac{73}{112}\right)\) | \(e\left(\frac{111}{112}\right)\) | \(e\left(\frac{121}{224}\right)\) | \(e\left(\frac{51}{224}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{211}{224}\right)\) | \(e\left(\frac{33}{224}\right)\) |
| \(\chi_{3712}(211,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{155}{224}\right)\) | \(e\left(\frac{129}{224}\right)\) | \(e\left(\frac{109}{112}\right)\) | \(e\left(\frac{43}{112}\right)\) | \(e\left(\frac{173}{224}\right)\) | \(e\left(\frac{47}{224}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{111}{224}\right)\) | \(e\left(\frac{149}{224}\right)\) |
| \(\chi_{3712}(235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{224}\right)\) | \(e\left(\frac{75}{224}\right)\) | \(e\left(\frac{79}{112}\right)\) | \(e\left(\frac{25}{112}\right)\) | \(e\left(\frac{111}{224}\right)\) | \(e\left(\frac{69}{224}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{101}{224}\right)\) | \(e\left(\frac{71}{224}\right)\) |
| \(\chi_{3712}(251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{224}\right)\) | \(e\left(\frac{23}{224}\right)\) | \(e\left(\frac{75}{112}\right)\) | \(e\left(\frac{45}{112}\right)\) | \(e\left(\frac{43}{224}\right)\) | \(e\left(\frac{57}{224}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{25}{224}\right)\) | \(e\left(\frac{195}{224}\right)\) |
| \(\chi_{3712}(259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{224}\right)\) | \(e\left(\frac{197}{224}\right)\) | \(e\left(\frac{97}{112}\right)\) | \(e\left(\frac{103}{112}\right)\) | \(e\left(\frac{193}{224}\right)\) | \(e\left(\frac{11}{224}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{107}{224}\right)\) | \(e\left(\frac{73}{224}\right)\) |
| \(\chi_{3712}(379,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{224}\right)\) | \(e\left(\frac{183}{224}\right)\) | \(e\left(\frac{27}{112}\right)\) | \(e\left(\frac{61}{112}\right)\) | \(e\left(\frac{11}{224}\right)\) | \(e\left(\frac{25}{224}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{121}{224}\right)\) | \(e\left(\frac{3}{224}\right)\) |
| \(\chi_{3712}(387,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{224}\right)\) | \(e\left(\frac{37}{224}\right)\) | \(e\left(\frac{33}{112}\right)\) | \(e\left(\frac{87}{112}\right)\) | \(e\left(\frac{1}{224}\right)\) | \(e\left(\frac{43}{224}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{224}\right)\) | \(e\left(\frac{41}{224}\right)\) |
| \(\chi_{3712}(403,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{224}\right)\) | \(e\left(\frac{145}{224}\right)\) | \(e\left(\frac{93}{112}\right)\) | \(e\left(\frac{11}{112}\right)\) | \(e\left(\frac{125}{224}\right)\) | \(e\left(\frac{223}{224}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{31}{224}\right)\) | \(e\left(\frac{197}{224}\right)\) |
| \(\chi_{3712}(427,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{224}\right)\) | \(e\left(\frac{59}{224}\right)\) | \(e\left(\frac{95}{112}\right)\) | \(e\left(\frac{57}{112}\right)\) | \(e\left(\frac{159}{224}\right)\) | \(e\left(\frac{117}{224}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{181}{224}\right)\) | \(e\left(\frac{23}{224}\right)\) |
| \(\chi_{3712}(475,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{224}\right)\) | \(e\left(\frac{95}{224}\right)\) | \(e\left(\frac{3}{112}\right)\) | \(e\left(\frac{69}{112}\right)\) | \(e\left(\frac{51}{224}\right)\) | \(e\left(\frac{177}{224}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{113}{224}\right)\) | \(e\left(\frac{75}{224}\right)\) |
| \(\chi_{3712}(507,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{205}{224}\right)\) | \(e\left(\frac{55}{224}\right)\) | \(e\left(\frac{43}{112}\right)\) | \(e\left(\frac{93}{112}\right)\) | \(e\left(\frac{171}{224}\right)\) | \(e\left(\frac{185}{224}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{89}{224}\right)\) | \(e\left(\frac{67}{224}\right)\) |
| \(\chi_{3712}(595,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{219}{224}\right)\) | \(e\left(\frac{97}{224}\right)\) | \(e\left(\frac{29}{112}\right)\) | \(e\left(\frac{107}{112}\right)\) | \(e\left(\frac{45}{224}\right)\) | \(e\left(\frac{143}{224}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{47}{224}\right)\) | \(e\left(\frac{53}{224}\right)\) |
| \(\chi_{3712}(627,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{195}{224}\right)\) | \(e\left(\frac{25}{224}\right)\) | \(e\left(\frac{101}{112}\right)\) | \(e\left(\frac{83}{112}\right)\) | \(e\left(\frac{37}{224}\right)\) | \(e\left(\frac{23}{224}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{183}{224}\right)\) | \(e\left(\frac{173}{224}\right)\) |
| \(\chi_{3712}(675,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{224}\right)\) | \(e\left(\frac{157}{224}\right)\) | \(e\left(\frac{25}{112}\right)\) | \(e\left(\frac{15}{112}\right)\) | \(e\left(\frac{89}{224}\right)\) | \(e\left(\frac{19}{224}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{83}{224}\right)\) | \(e\left(\frac{65}{224}\right)\) |
| \(\chi_{3712}(699,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{221}{224}\right)\) | \(e\left(\frac{103}{224}\right)\) | \(e\left(\frac{107}{112}\right)\) | \(e\left(\frac{109}{112}\right)\) | \(e\left(\frac{27}{224}\right)\) | \(e\left(\frac{41}{224}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{73}{224}\right)\) | \(e\left(\frac{211}{224}\right)\) |
| \(\chi_{3712}(715,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{129}{224}\right)\) | \(e\left(\frac{51}{224}\right)\) | \(e\left(\frac{103}{112}\right)\) | \(e\left(\frac{17}{112}\right)\) | \(e\left(\frac{183}{224}\right)\) | \(e\left(\frac{29}{224}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{221}{224}\right)\) | \(e\left(\frac{111}{224}\right)\) |
| \(\chi_{3712}(723,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{187}{224}\right)\) | \(e\left(\frac{1}{224}\right)\) | \(e\left(\frac{13}{112}\right)\) | \(e\left(\frac{75}{112}\right)\) | \(e\left(\frac{109}{224}\right)\) | \(e\left(\frac{207}{224}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{79}{224}\right)\) | \(e\left(\frac{213}{224}\right)\) |
| \(\chi_{3712}(843,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{224}\right)\) | \(e\left(\frac{211}{224}\right)\) | \(e\left(\frac{55}{112}\right)\) | \(e\left(\frac{33}{112}\right)\) | \(e\left(\frac{151}{224}\right)\) | \(e\left(\frac{221}{224}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{93}{224}\right)\) | \(e\left(\frac{143}{224}\right)\) |
| \(\chi_{3712}(851,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{224}\right)\) | \(e\left(\frac{65}{224}\right)\) | \(e\left(\frac{61}{112}\right)\) | \(e\left(\frac{59}{112}\right)\) | \(e\left(\frac{141}{224}\right)\) | \(e\left(\frac{15}{224}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{207}{224}\right)\) | \(e\left(\frac{181}{224}\right)\) |
| \(\chi_{3712}(867,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{224}\right)\) | \(e\left(\frac{173}{224}\right)\) | \(e\left(\frac{9}{112}\right)\) | \(e\left(\frac{95}{112}\right)\) | \(e\left(\frac{41}{224}\right)\) | \(e\left(\frac{195}{224}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{224}\right)\) | \(e\left(\frac{113}{224}\right)\) |
| \(\chi_{3712}(891,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{141}{224}\right)\) | \(e\left(\frac{87}{224}\right)\) | \(e\left(\frac{11}{112}\right)\) | \(e\left(\frac{29}{112}\right)\) | \(e\left(\frac{75}{224}\right)\) | \(e\left(\frac{89}{224}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{153}{224}\right)\) | \(e\left(\frac{163}{224}\right)\) |
| \(\chi_{3712}(939,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{153}{224}\right)\) | \(e\left(\frac{123}{224}\right)\) | \(e\left(\frac{31}{112}\right)\) | \(e\left(\frac{41}{112}\right)\) | \(e\left(\frac{191}{224}\right)\) | \(e\left(\frac{149}{224}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{85}{224}\right)\) | \(e\left(\frac{215}{224}\right)\) |
| \(\chi_{3712}(971,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{224}\right)\) | \(e\left(\frac{83}{224}\right)\) | \(e\left(\frac{71}{112}\right)\) | \(e\left(\frac{65}{112}\right)\) | \(e\left(\frac{87}{224}\right)\) | \(e\left(\frac{157}{224}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{61}{224}\right)\) | \(e\left(\frac{207}{224}\right)\) |
| \(\chi_{3712}(1059,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{224}\right)\) | \(e\left(\frac{125}{224}\right)\) | \(e\left(\frac{57}{112}\right)\) | \(e\left(\frac{79}{112}\right)\) | \(e\left(\frac{185}{224}\right)\) | \(e\left(\frac{115}{224}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{224}\right)\) | \(e\left(\frac{193}{224}\right)\) |
| \(\chi_{3712}(1091,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{224}\right)\) | \(e\left(\frac{53}{224}\right)\) | \(e\left(\frac{17}{112}\right)\) | \(e\left(\frac{55}{112}\right)\) | \(e\left(\frac{177}{224}\right)\) | \(e\left(\frac{219}{224}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{155}{224}\right)\) | \(e\left(\frac{89}{224}\right)\) |
| \(\chi_{3712}(1139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{99}{224}\right)\) | \(e\left(\frac{185}{224}\right)\) | \(e\left(\frac{53}{112}\right)\) | \(e\left(\frac{99}{112}\right)\) | \(e\left(\frac{5}{224}\right)\) | \(e\left(\frac{215}{224}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{55}{224}\right)\) | \(e\left(\frac{205}{224}\right)\) |
| \(\chi_{3712}(1163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{224}\right)\) | \(e\left(\frac{131}{224}\right)\) | \(e\left(\frac{23}{112}\right)\) | \(e\left(\frac{81}{112}\right)\) | \(e\left(\frac{167}{224}\right)\) | \(e\left(\frac{13}{224}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{45}{224}\right)\) | \(e\left(\frac{127}{224}\right)\) |
| \(\chi_{3712}(1179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{213}{224}\right)\) | \(e\left(\frac{79}{224}\right)\) | \(e\left(\frac{19}{112}\right)\) | \(e\left(\frac{101}{112}\right)\) | \(e\left(\frac{99}{224}\right)\) | \(e\left(\frac{1}{224}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{193}{224}\right)\) | \(e\left(\frac{27}{224}\right)\) |