from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(113, base_ring=CyclotomicField(112))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,113))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(113\) | |
Conductor: | \(113\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(112\) | 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: | $\Q(\zeta_{112})$ |
Fixed field: | Number field defined by a degree 112 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{113}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{112}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{83}{112}\right)\) | \(e\left(\frac{13}{112}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{95}{112}\right)\) | \(e\left(\frac{37}{56}\right)\) |
\(\chi_{113}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{83}{112}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{57}{112}\right)\) | \(e\left(\frac{71}{112}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{45}{112}\right)\) | \(e\left(\frac{47}{56}\right)\) |
\(\chi_{113}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{13}{112}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{71}{112}\right)\) | \(e\left(\frac{57}{112}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{3}{112}\right)\) | \(e\left(\frac{33}{56}\right)\) |
\(\chi_{113}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{95}{112}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{45}{112}\right)\) | \(e\left(\frac{3}{112}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{65}{112}\right)\) | \(e\left(\frac{43}{56}\right)\) |
\(\chi_{113}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{25}{112}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{59}{112}\right)\) | \(e\left(\frac{101}{112}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{23}{112}\right)\) | \(e\left(\frac{29}{56}\right)\) |
\(\chi_{113}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{112}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{79}{112}\right)\) | \(e\left(\frac{65}{112}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{27}{112}\right)\) | \(e\left(\frac{17}{56}\right)\) |
\(\chi_{113}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{99}{112}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{41}{112}\right)\) | \(e\left(\frac{55}{112}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{109}{112}\right)\) | \(e\left(\frac{23}{56}\right)\) |
\(\chi_{113}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{107}{112}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{33}{112}\right)\) | \(e\left(\frac{47}{112}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{85}{112}\right)\) | \(e\left(\frac{39}{56}\right)\) |
\(\chi_{113}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{9}{112}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{75}{112}\right)\) | \(e\left(\frac{5}{112}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{71}{112}\right)\) | \(e\left(\frac{53}{56}\right)\) |
\(\chi_{113}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{41}{112}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{43}{112}\right)\) | \(e\left(\frac{85}{112}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{87}{112}\right)\) | \(e\left(\frac{5}{56}\right)\) |
\(\chi_{113}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{37}{112}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{47}{112}\right)\) | \(e\left(\frac{33}{112}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{43}{112}\right)\) | \(e\left(\frac{25}{56}\right)\) |
\(\chi_{113}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{112}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{25}{112}\right)\) | \(e\left(\frac{39}{112}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{61}{112}\right)\) | \(e\left(\frac{55}{56}\right)\) |
\(\chi_{113}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{89}{112}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{107}{112}\right)\) | \(e\left(\frac{37}{112}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{55}{112}\right)\) | \(e\left(\frac{45}{56}\right)\) |
\(\chi_{113}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{75}{112}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{65}{112}\right)\) | \(e\left(\frac{79}{112}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{69}{112}\right)\) | \(e\left(\frac{31}{56}\right)\) |
\(\chi_{113}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{17}{112}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{67}{112}\right)\) | \(e\left(\frac{109}{112}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{47}{112}\right)\) | \(e\left(\frac{13}{56}\right)\) |
\(\chi_{113}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{67}{112}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{73}{112}\right)\) | \(e\left(\frac{87}{112}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{93}{112}\right)\) | \(e\left(\frac{15}{56}\right)\) |
\(\chi_{113}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{111}{112}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{29}{112}\right)\) | \(e\left(\frac{99}{112}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{17}{112}\right)\) | \(e\left(\frac{19}{56}\right)\) |
\(\chi_{113}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{23}{112}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{112}\right)\) | \(e\left(\frac{75}{112}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{57}{112}\right)\) | \(e\left(\frac{11}{56}\right)\) |
\(\chi_{113}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{47}{112}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{93}{112}\right)\) | \(e\left(\frac{51}{112}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{97}{112}\right)\) | \(e\left(\frac{3}{56}\right)\) |
\(\chi_{113}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{85}{112}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{111}{112}\right)\) | \(e\left(\frac{97}{112}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{11}{112}\right)\) | \(e\left(\frac{9}{56}\right)\) |
\(\chi_{113}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{53}{112}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{31}{112}\right)\) | \(e\left(\frac{17}{112}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{107}{112}\right)\) | \(e\left(\frac{1}{56}\right)\) |
\(\chi_{113}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{31}{112}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{109}{112}\right)\) | \(e\left(\frac{67}{112}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{33}{112}\right)\) | \(e\left(\frac{27}{56}\right)\) |
\(\chi_{113}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{15}{112}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{112}\right)\) | \(e\left(\frac{83}{112}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{81}{112}\right)\) | \(e\left(\frac{51}{56}\right)\) |
\(\chi_{113}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{45}{112}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{39}{112}\right)\) | \(e\left(\frac{25}{112}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{19}{112}\right)\) | \(e\left(\frac{41}{56}\right)\) |
\(\chi_{113}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{101}{112}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{95}{112}\right)\) | \(e\left(\frac{81}{112}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{75}{112}\right)\) | \(e\left(\frac{41}{56}\right)\) |
\(\chi_{113}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{71}{112}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{69}{112}\right)\) | \(e\left(\frac{27}{112}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{25}{112}\right)\) | \(e\left(\frac{51}{56}\right)\) |
\(\chi_{113}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{87}{112}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{53}{112}\right)\) | \(e\left(\frac{11}{112}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{89}{112}\right)\) | \(e\left(\frac{27}{56}\right)\) |
\(\chi_{113}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{109}{112}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{87}{112}\right)\) | \(e\left(\frac{73}{112}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{51}{112}\right)\) | \(e\left(\frac{1}{56}\right)\) |
\(\chi_{113}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{29}{112}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{55}{112}\right)\) | \(e\left(\frac{41}{112}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{67}{112}\right)\) | \(e\left(\frac{9}{56}\right)\) |
\(\chi_{113}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{103}{112}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{37}{112}\right)\) | \(e\left(\frac{107}{112}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{41}{112}\right)\) | \(e\left(\frac{3}{56}\right)\) |
\(\chi_{113}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{79}{112}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{61}{112}\right)\) | \(e\left(\frac{19}{112}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{1}{112}\right)\) | \(e\left(\frac{11}{56}\right)\) |