Properties

Label 712.bf
Modulus $712$
Conductor $712$
Order $88$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(712, base_ring=CyclotomicField(88))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([44,44,1]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(3,712))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(712\)
Conductor: \(712\)
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: even
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 \(-1\) \(1\) \(3\) \(5\) \(7\) \(9\) \(11\) \(13\) \(15\) \(17\) \(19\) \(21\)
\(\chi_{712}(3,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{88}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{37}{88}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{67}{88}\right)\) \(e\left(\frac{27}{88}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{35}{88}\right)\) \(e\left(\frac{19}{44}\right)\)
\(\chi_{712}(19,\cdot)\) \(1\) \(1\) \(e\left(\frac{35}{88}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{63}{88}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{57}{88}\right)\) \(e\left(\frac{65}{88}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{81}{88}\right)\) \(e\left(\frac{5}{44}\right)\)
\(\chi_{712}(27,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{88}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{23}{88}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{25}{88}\right)\) \(e\left(\frac{81}{88}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{17}{88}\right)\) \(e\left(\frac{13}{44}\right)\)
\(\chi_{712}(35,\cdot)\) \(1\) \(1\) \(e\left(\frac{63}{88}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{43}{88}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{85}{88}\right)\) \(e\left(\frac{29}{88}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{5}{88}\right)\) \(e\left(\frac{9}{44}\right)\)
\(\chi_{712}(43,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{88}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{17}{88}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{7}{88}\right)\) \(e\left(\frac{79}{88}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{47}{88}\right)\) \(e\left(\frac{23}{44}\right)\)
\(\chi_{712}(51,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{88}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{83}{88}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{29}{88}\right)\) \(e\left(\frac{13}{88}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{69}{88}\right)\) \(e\left(\frac{1}{44}\right)\)
\(\chi_{712}(59,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{88}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{7}{88}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{65}{88}\right)\) \(e\left(\frac{17}{88}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{9}{88}\right)\) \(e\left(\frac{25}{44}\right)\)
\(\chi_{712}(75,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{88}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{25}{88}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{31}{88}\right)\) \(e\left(\frac{23}{88}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{7}{88}\right)\) \(e\left(\frac{39}{44}\right)\)
\(\chi_{712}(83,\cdot)\) \(1\) \(1\) \(e\left(\frac{61}{88}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{57}{88}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{39}{88}\right)\) \(e\left(\frac{63}{88}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{23}{88}\right)\) \(e\left(\frac{15}{44}\right)\)
\(\chi_{712}(115,\cdot)\) \(1\) \(1\) \(e\left(\frac{39}{88}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{35}{88}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{61}{88}\right)\) \(e\left(\frac{85}{88}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{45}{88}\right)\) \(e\left(\frac{37}{44}\right)\)
\(\chi_{712}(147,\cdot)\) \(1\) \(1\) \(e\left(\frac{75}{88}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{47}{88}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{9}{88}\right)\) \(e\left(\frac{1}{88}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{73}{88}\right)\) \(e\left(\frac{17}{44}\right)\)
\(\chi_{712}(155,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{88}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{41}{88}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{79}{88}\right)\) \(e\left(\frac{87}{88}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{15}{88}\right)\) \(e\left(\frac{27}{44}\right)\)
\(\chi_{712}(163,\cdot)\) \(1\) \(1\) \(e\left(\frac{27}{88}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{31}{88}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{49}{88}\right)\) \(e\left(\frac{25}{88}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{65}{88}\right)\) \(e\left(\frac{29}{44}\right)\)
\(\chi_{712}(171,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{88}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{49}{88}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{15}{88}\right)\) \(e\left(\frac{31}{88}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{63}{88}\right)\) \(e\left(\frac{43}{44}\right)\)
\(\chi_{712}(211,\cdot)\) \(1\) \(1\) \(e\left(\frac{85}{88}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{65}{88}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{63}{88}\right)\) \(e\left(\frac{7}{88}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{71}{88}\right)\) \(e\left(\frac{31}{44}\right)\)
\(\chi_{712}(219,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{88}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{73}{88}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{87}{88}\right)\) \(e\left(\frac{39}{88}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{31}{88}\right)\) \(e\left(\frac{3}{44}\right)\)
\(\chi_{712}(243,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{88}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{9}{88}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{71}{88}\right)\) \(e\left(\frac{47}{88}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{87}{88}\right)\) \(e\left(\frac{7}{44}\right)\)
\(\chi_{712}(291,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{88}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{53}{88}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{27}{88}\right)\) \(e\left(\frac{3}{88}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{43}{88}\right)\) \(e\left(\frac{7}{44}\right)\)
\(\chi_{712}(315,\cdot)\) \(1\) \(1\) \(e\left(\frac{65}{88}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{29}{88}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{43}{88}\right)\) \(e\left(\frac{83}{88}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{75}{88}\right)\) \(e\left(\frac{3}{44}\right)\)
\(\chi_{712}(323,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{88}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{21}{88}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{19}{88}\right)\) \(e\left(\frac{51}{88}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{27}{88}\right)\) \(e\left(\frac{31}{44}\right)\)
\(\chi_{712}(363,\cdot)\) \(1\) \(1\) \(e\left(\frac{81}{88}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{5}{88}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{59}{88}\right)\) \(e\left(\frac{75}{88}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{19}{88}\right)\) \(e\left(\frac{43}{44}\right)\)
\(\chi_{712}(371,\cdot)\) \(1\) \(1\) \(e\left(\frac{71}{88}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{75}{88}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{5}{88}\right)\) \(e\left(\frac{69}{88}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{21}{88}\right)\) \(e\left(\frac{29}{44}\right)\)
\(\chi_{712}(379,\cdot)\) \(1\) \(1\) \(e\left(\frac{57}{88}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{85}{88}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{35}{88}\right)\) \(e\left(\frac{43}{88}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{59}{88}\right)\) \(e\left(\frac{27}{44}\right)\)
\(\chi_{712}(387,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{88}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{3}{88}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{53}{88}\right)\) \(e\left(\frac{45}{88}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{29}{88}\right)\) \(e\left(\frac{17}{44}\right)\)
\(\chi_{712}(419,\cdot)\) \(1\) \(1\) \(e\left(\frac{83}{88}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{79}{88}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{17}{88}\right)\) \(e\left(\frac{41}{88}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{1}{88}\right)\) \(e\left(\frac{37}{44}\right)\)
\(\chi_{712}(451,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{88}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{13}{88}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{83}{88}\right)\) \(e\left(\frac{19}{88}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{67}{88}\right)\) \(e\left(\frac{15}{44}\right)\)
\(\chi_{712}(459,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{88}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{69}{88}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{75}{88}\right)\) \(e\left(\frac{67}{88}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{51}{88}\right)\) \(e\left(\frac{39}{44}\right)\)
\(\chi_{712}(475,\cdot)\) \(1\) \(1\) \(e\left(\frac{87}{88}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{51}{88}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{21}{88}\right)\) \(e\left(\frac{61}{88}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{53}{88}\right)\) \(e\left(\frac{25}{44}\right)\)
\(\chi_{712}(483,\cdot)\) \(1\) \(1\) \(e\left(\frac{51}{88}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{39}{88}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{73}{88}\right)\) \(e\left(\frac{57}{88}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{25}{88}\right)\) \(e\left(\frac{1}{44}\right)\)
\(\chi_{712}(491,\cdot)\) \(1\) \(1\) \(e\left(\frac{73}{88}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{61}{88}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{51}{88}\right)\) \(e\left(\frac{35}{88}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{3}{88}\right)\) \(e\left(\frac{23}{44}\right)\)
\(\chi_{712}(499,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{88}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{87}{88}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{41}{88}\right)\) \(e\left(\frac{73}{88}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{49}{88}\right)\) \(e\left(\frac{9}{44}\right)\)