Properties

Label 89.h
Modulus $89$
Conductor $89$
Order $88$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(89, base_ring=CyclotomicField(88))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(3,89))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(89\)
Conductor: \(89\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(88\)
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: $\Q(\zeta_{88})$
Fixed field: Number field defined by a degree 88 polynomial

First 31 of 40 characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{89}(3,\cdot)\) \(-1\) \(1\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{1}{88}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{17}{88}\right)\) \(e\left(\frac{81}{88}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{21}{22}\right)\)
\(\chi_{89}(6,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{17}{88}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{25}{88}\right)\) \(e\left(\frac{57}{88}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{5}{22}\right)\)
\(\chi_{89}(7,\cdot)\) \(-1\) \(1\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{81}{88}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{57}{88}\right)\) \(e\left(\frac{49}{88}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{7}{22}\right)\)
\(\chi_{89}(13,\cdot)\) \(-1\) \(1\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{23}{88}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{39}{88}\right)\) \(e\left(\frac{15}{88}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{21}{22}\right)\)
\(\chi_{89}(14,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{9}{88}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{65}{88}\right)\) \(e\left(\frac{25}{88}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{13}{22}\right)\)
\(\chi_{89}(15,\cdot)\) \(-1\) \(1\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{71}{88}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{63}{88}\right)\) \(e\left(\frac{31}{88}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{17}{22}\right)\)
\(\chi_{89}(19,\cdot)\) \(-1\) \(1\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{35}{88}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{67}{88}\right)\) \(e\left(\frac{19}{88}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{9}{22}\right)\)
\(\chi_{89}(23,\cdot)\) \(-1\) \(1\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{57}{88}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{1}{88}\right)\) \(e\left(\frac{41}{88}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{9}{22}\right)\)
\(\chi_{89}(24,\cdot)\) \(-1\) \(1\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{49}{88}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{41}{88}\right)\) \(e\left(\frac{9}{88}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{17}{22}\right)\)
\(\chi_{89}(26,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{39}{88}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{47}{88}\right)\) \(e\left(\frac{79}{88}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{5}{22}\right)\)
\(\chi_{89}(27,\cdot)\) \(-1\) \(1\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{3}{88}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{51}{88}\right)\) \(e\left(\frac{67}{88}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{19}{22}\right)\)
\(\chi_{89}(28,\cdot)\) \(-1\) \(1\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{25}{88}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{73}{88}\right)\) \(e\left(\frac{1}{88}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{19}{22}\right)\)
\(\chi_{89}(29,\cdot)\) \(-1\) \(1\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{59}{88}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{35}{88}\right)\) \(e\left(\frac{27}{88}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{7}{22}\right)\)
\(\chi_{89}(30,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{87}{88}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{71}{88}\right)\) \(e\left(\frac{7}{88}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{1}{22}\right)\)
\(\chi_{89}(31,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{31}{88}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{87}{88}\right)\) \(e\left(\frac{47}{88}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{13}{22}\right)\)
\(\chi_{89}(33,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{85}{88}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{37}{88}\right)\) \(e\left(\frac{21}{88}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{3}{22}\right)\)
\(\chi_{89}(35,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{63}{88}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{15}{88}\right)\) \(e\left(\frac{87}{88}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{3}{22}\right)\)
\(\chi_{89}(38,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{51}{88}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{75}{88}\right)\) \(e\left(\frac{83}{88}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{15}{22}\right)\)
\(\chi_{89}(41,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{21}{88}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{5}{88}\right)\) \(e\left(\frac{29}{88}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{1}{22}\right)\)
\(\chi_{89}(43,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{29}{88}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{53}{88}\right)\) \(e\left(\frac{61}{88}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{15}{22}\right)\)
\(\chi_{89}(46,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{73}{88}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{9}{88}\right)\) \(e\left(\frac{17}{88}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{15}{22}\right)\)
\(\chi_{89}(48,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{65}{88}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{49}{88}\right)\) \(e\left(\frac{73}{88}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{1}{22}\right)\)
\(\chi_{89}(51,\cdot)\) \(-1\) \(1\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{7}{88}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{31}{88}\right)\) \(e\left(\frac{39}{88}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{15}{22}\right)\)
\(\chi_{89}(54,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{19}{88}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{59}{88}\right)\) \(e\left(\frac{43}{88}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{3}{22}\right)\)
\(\chi_{89}(56,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{41}{88}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{81}{88}\right)\) \(e\left(\frac{65}{88}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{3}{22}\right)\)
\(\chi_{89}(58,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{75}{88}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{43}{88}\right)\) \(e\left(\frac{3}{88}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{13}{22}\right)\)
\(\chi_{89}(59,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{43}{88}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{27}{88}\right)\) \(e\left(\frac{51}{88}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{1}{22}\right)\)
\(\chi_{89}(60,\cdot)\) \(-1\) \(1\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{15}{88}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{79}{88}\right)\) \(e\left(\frac{71}{88}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{7}{22}\right)\)
\(\chi_{89}(61,\cdot)\) \(-1\) \(1\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{69}{88}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{29}{88}\right)\) \(e\left(\frac{45}{88}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{19}{22}\right)\)
\(\chi_{89}(62,\cdot)\) \(-1\) \(1\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{47}{88}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{7}{88}\right)\) \(e\left(\frac{23}{88}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{19}{22}\right)\)
\(\chi_{89}(63,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{83}{88}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{3}{88}\right)\) \(e\left(\frac{35}{88}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{5}{22}\right)\)