Properties

Label 113.j
Modulus $113$
Conductor $113$
Order $112$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
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 \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{113}(3,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{1}{112}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{83}{112}\right)\) \(e\left(\frac{13}{112}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{1}{56}\right)\) \(e\left(\frac{95}{112}\right)\) \(e\left(\frac{37}{56}\right)\)
\(\chi_{113}(5,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{83}{112}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{57}{112}\right)\) \(e\left(\frac{71}{112}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{27}{56}\right)\) \(e\left(\frac{45}{112}\right)\) \(e\left(\frac{47}{56}\right)\)
\(\chi_{113}(6,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{13}{112}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{71}{112}\right)\) \(e\left(\frac{57}{112}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{13}{56}\right)\) \(e\left(\frac{3}{112}\right)\) \(e\left(\frac{33}{56}\right)\)
\(\chi_{113}(10,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{95}{112}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{45}{112}\right)\) \(e\left(\frac{3}{112}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{39}{56}\right)\) \(e\left(\frac{65}{112}\right)\) \(e\left(\frac{43}{56}\right)\)
\(\chi_{113}(12,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{25}{112}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{59}{112}\right)\) \(e\left(\frac{101}{112}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{25}{56}\right)\) \(e\left(\frac{23}{112}\right)\) \(e\left(\frac{29}{56}\right)\)
\(\chi_{113}(17,\cdot)\) \(-1\) \(1\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{5}{112}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{79}{112}\right)\) \(e\left(\frac{65}{112}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{5}{56}\right)\) \(e\left(\frac{27}{112}\right)\) \(e\left(\frac{17}{56}\right)\)
\(\chi_{113}(19,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{99}{112}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{41}{112}\right)\) \(e\left(\frac{55}{112}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{43}{56}\right)\) \(e\left(\frac{109}{112}\right)\) \(e\left(\frac{23}{56}\right)\)
\(\chi_{113}(20,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{107}{112}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{33}{112}\right)\) \(e\left(\frac{47}{112}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{51}{56}\right)\) \(e\left(\frac{85}{112}\right)\) \(e\left(\frac{39}{56}\right)\)
\(\chi_{113}(21,\cdot)\) \(-1\) \(1\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{9}{112}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{75}{112}\right)\) \(e\left(\frac{5}{112}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{9}{56}\right)\) \(e\left(\frac{71}{112}\right)\) \(e\left(\frac{53}{56}\right)\)
\(\chi_{113}(23,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{41}{112}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{43}{112}\right)\) \(e\left(\frac{85}{112}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{41}{56}\right)\) \(e\left(\frac{87}{112}\right)\) \(e\left(\frac{5}{56}\right)\)
\(\chi_{113}(24,\cdot)\) \(-1\) \(1\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{37}{112}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{47}{112}\right)\) \(e\left(\frac{33}{112}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{37}{56}\right)\) \(e\left(\frac{43}{112}\right)\) \(e\left(\frac{25}{56}\right)\)
\(\chi_{113}(27,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{3}{112}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{25}{112}\right)\) \(e\left(\frac{39}{112}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{3}{56}\right)\) \(e\left(\frac{61}{112}\right)\) \(e\left(\frac{55}{56}\right)\)
\(\chi_{113}(29,\cdot)\) \(-1\) \(1\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{89}{112}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{107}{112}\right)\) \(e\left(\frac{37}{112}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{33}{56}\right)\) \(e\left(\frac{55}{112}\right)\) \(e\left(\frac{45}{56}\right)\)
\(\chi_{113}(33,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{75}{112}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{65}{112}\right)\) \(e\left(\frac{79}{112}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{19}{56}\right)\) \(e\left(\frac{69}{112}\right)\) \(e\left(\frac{31}{56}\right)\)
\(\chi_{113}(34,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{17}{112}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{67}{112}\right)\) \(e\left(\frac{109}{112}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{17}{56}\right)\) \(e\left(\frac{47}{112}\right)\) \(e\left(\frac{13}{56}\right)\)
\(\chi_{113}(37,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{67}{112}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{73}{112}\right)\) \(e\left(\frac{87}{112}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{11}{56}\right)\) \(e\left(\frac{93}{112}\right)\) \(e\left(\frac{15}{56}\right)\)
\(\chi_{113}(38,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{111}{112}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{29}{112}\right)\) \(e\left(\frac{99}{112}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{55}{56}\right)\) \(e\left(\frac{17}{112}\right)\) \(e\left(\frac{19}{56}\right)\)
\(\chi_{113}(39,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{23}{112}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{5}{112}\right)\) \(e\left(\frac{75}{112}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{23}{56}\right)\) \(e\left(\frac{57}{112}\right)\) \(e\left(\frac{11}{56}\right)\)
\(\chi_{113}(43,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{47}{112}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{93}{112}\right)\) \(e\left(\frac{51}{112}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{47}{56}\right)\) \(e\left(\frac{97}{112}\right)\) \(e\left(\frac{3}{56}\right)\)
\(\chi_{113}(45,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{85}{112}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{111}{112}\right)\) \(e\left(\frac{97}{112}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{29}{56}\right)\) \(e\left(\frac{11}{112}\right)\) \(e\left(\frac{9}{56}\right)\)
\(\chi_{113}(46,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{53}{112}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{31}{112}\right)\) \(e\left(\frac{17}{112}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{53}{56}\right)\) \(e\left(\frac{107}{112}\right)\) \(e\left(\frac{1}{56}\right)\)
\(\chi_{113}(47,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{31}{112}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{109}{112}\right)\) \(e\left(\frac{67}{112}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{31}{56}\right)\) \(e\left(\frac{33}{112}\right)\) \(e\left(\frac{27}{56}\right)\)
\(\chi_{113}(54,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{15}{112}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{13}{112}\right)\) \(e\left(\frac{83}{112}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{15}{56}\right)\) \(e\left(\frac{81}{112}\right)\) \(e\left(\frac{51}{56}\right)\)
\(\chi_{113}(55,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{45}{112}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{39}{112}\right)\) \(e\left(\frac{25}{112}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{45}{56}\right)\) \(e\left(\frac{19}{112}\right)\) \(e\left(\frac{41}{56}\right)\)
\(\chi_{113}(58,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{101}{112}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{95}{112}\right)\) \(e\left(\frac{81}{112}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{45}{56}\right)\) \(e\left(\frac{75}{112}\right)\) \(e\left(\frac{41}{56}\right)\)
\(\chi_{113}(59,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{71}{112}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{69}{112}\right)\) \(e\left(\frac{27}{112}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{15}{56}\right)\) \(e\left(\frac{25}{112}\right)\) \(e\left(\frac{51}{56}\right)\)
\(\chi_{113}(66,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{87}{112}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{53}{112}\right)\) \(e\left(\frac{11}{112}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{31}{56}\right)\) \(e\left(\frac{89}{112}\right)\) \(e\left(\frac{27}{56}\right)\)
\(\chi_{113}(67,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{109}{112}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{87}{112}\right)\) \(e\left(\frac{73}{112}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{53}{56}\right)\) \(e\left(\frac{51}{112}\right)\) \(e\left(\frac{1}{56}\right)\)
\(\chi_{113}(68,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{29}{112}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{55}{112}\right)\) \(e\left(\frac{41}{112}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{29}{56}\right)\) \(e\left(\frac{67}{112}\right)\) \(e\left(\frac{9}{56}\right)\)
\(\chi_{113}(70,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{103}{112}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{37}{112}\right)\) \(e\left(\frac{107}{112}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{47}{56}\right)\) \(e\left(\frac{41}{112}\right)\) \(e\left(\frac{3}{56}\right)\)
\(\chi_{113}(74,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{79}{112}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{61}{112}\right)\) \(e\left(\frac{19}{112}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{23}{56}\right)\) \(e\left(\frac{1}{112}\right)\) \(e\left(\frac{11}{56}\right)\)