Properties

Label 712.bc
Modulus $712$
Conductor $712$
Order $88$
Real no
Primitive yes
Minimal yes
Parity odd

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([0,44,23]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(13,712))
 
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: 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 \(-1\) \(1\) \(3\) \(5\) \(7\) \(9\) \(11\) \(13\) \(15\) \(17\) \(19\) \(21\)
\(\chi_{712}(13,\cdot)\) \(-1\) \(1\) \(e\left(\frac{67}{88}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{15}{88}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{45}{88}\right)\) \(e\left(\frac{49}{88}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{57}{88}\right)\) \(e\left(\frac{41}{44}\right)\)
\(\chi_{712}(29,\cdot)\) \(-1\) \(1\) \(e\left(\frac{15}{88}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{27}{88}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{81}{88}\right)\) \(e\left(\frac{53}{88}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{85}{88}\right)\) \(e\left(\frac{21}{44}\right)\)
\(\chi_{712}(61,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{88}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{45}{88}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{47}{88}\right)\) \(e\left(\frac{59}{88}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{83}{88}\right)\) \(e\left(\frac{35}{44}\right)\)
\(\chi_{712}(117,\cdot)\) \(-1\) \(1\) \(e\left(\frac{69}{88}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{1}{88}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{3}{88}\right)\) \(e\left(\frac{15}{88}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{39}{88}\right)\) \(e\left(\frac{35}{44}\right)\)
\(\chi_{712}(149,\cdot)\) \(-1\) \(1\) \(e\left(\frac{59}{88}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{71}{88}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{37}{88}\right)\) \(e\left(\frac{9}{88}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{41}{88}\right)\) \(e\left(\frac{21}{44}\right)\)
\(\chi_{712}(165,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{88}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{59}{88}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{1}{88}\right)\) \(e\left(\frac{5}{88}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{13}{88}\right)\) \(e\left(\frac{41}{44}\right)\)
\(\chi_{712}(181,\cdot)\) \(-1\) \(1\) \(e\left(\frac{45}{88}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{81}{88}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{67}{88}\right)\) \(e\left(\frac{71}{88}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{79}{88}\right)\) \(e\left(\frac{19}{44}\right)\)
\(\chi_{712}(197,\cdot)\) \(-1\) \(1\) \(e\left(\frac{79}{88}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{19}{88}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{57}{88}\right)\) \(e\left(\frac{21}{88}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{37}{88}\right)\) \(e\left(\frac{5}{44}\right)\)
\(\chi_{712}(205,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{88}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{67}{88}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{25}{88}\right)\) \(e\left(\frac{37}{88}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{61}{88}\right)\) \(e\left(\frac{13}{44}\right)\)
\(\chi_{712}(213,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{88}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{87}{88}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{85}{88}\right)\) \(e\left(\frac{73}{88}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{49}{88}\right)\) \(e\left(\frac{9}{44}\right)\)
\(\chi_{712}(221,\cdot)\) \(-1\) \(1\) \(e\left(\frac{73}{88}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{61}{88}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{7}{88}\right)\) \(e\left(\frac{35}{88}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{3}{88}\right)\) \(e\left(\frac{23}{44}\right)\)
\(\chi_{712}(229,\cdot)\) \(-1\) \(1\) \(e\left(\frac{51}{88}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{39}{88}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{29}{88}\right)\) \(e\left(\frac{57}{88}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{25}{88}\right)\) \(e\left(\frac{1}{44}\right)\)
\(\chi_{712}(237,\cdot)\) \(-1\) \(1\) \(e\left(\frac{87}{88}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{51}{88}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{65}{88}\right)\) \(e\left(\frac{61}{88}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{53}{88}\right)\) \(e\left(\frac{25}{44}\right)\)
\(\chi_{712}(253,\cdot)\) \(-1\) \(1\) \(e\left(\frac{9}{88}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{69}{88}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{31}{88}\right)\) \(e\left(\frac{67}{88}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{51}{88}\right)\) \(e\left(\frac{39}{44}\right)\)
\(\chi_{712}(261,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{88}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{13}{88}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{39}{88}\right)\) \(e\left(\frac{19}{88}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{67}{88}\right)\) \(e\left(\frac{15}{44}\right)\)
\(\chi_{712}(293,\cdot)\) \(-1\) \(1\) \(e\left(\frac{83}{88}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{79}{88}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{61}{88}\right)\) \(e\left(\frac{41}{88}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{1}{88}\right)\) \(e\left(\frac{37}{44}\right)\)
\(\chi_{712}(325,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{88}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{3}{88}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{9}{88}\right)\) \(e\left(\frac{45}{88}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{29}{88}\right)\) \(e\left(\frac{17}{44}\right)\)
\(\chi_{712}(333,\cdot)\) \(-1\) \(1\) \(e\left(\frac{57}{88}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{85}{88}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{79}{88}\right)\) \(e\left(\frac{43}{88}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{59}{88}\right)\) \(e\left(\frac{27}{44}\right)\)
\(\chi_{712}(341,\cdot)\) \(-1\) \(1\) \(e\left(\frac{71}{88}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{75}{88}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{49}{88}\right)\) \(e\left(\frac{69}{88}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{21}{88}\right)\) \(e\left(\frac{29}{44}\right)\)
\(\chi_{712}(349,\cdot)\) \(-1\) \(1\) \(e\left(\frac{81}{88}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{5}{88}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{15}{88}\right)\) \(e\left(\frac{75}{88}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{19}{88}\right)\) \(e\left(\frac{43}{44}\right)\)
\(\chi_{712}(389,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{88}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{21}{88}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{63}{88}\right)\) \(e\left(\frac{51}{88}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{27}{88}\right)\) \(e\left(\frac{31}{44}\right)\)
\(\chi_{712}(397,\cdot)\) \(-1\) \(1\) \(e\left(\frac{65}{88}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{29}{88}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{87}{88}\right)\) \(e\left(\frac{83}{88}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{75}{88}\right)\) \(e\left(\frac{3}{44}\right)\)
\(\chi_{712}(421,\cdot)\) \(-1\) \(1\) \(e\left(\frac{49}{88}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{53}{88}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{71}{88}\right)\) \(e\left(\frac{3}{88}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{43}{88}\right)\) \(e\left(\frac{7}{44}\right)\)
\(\chi_{712}(469,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{88}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{9}{88}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{27}{88}\right)\) \(e\left(\frac{47}{88}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{87}{88}\right)\) \(e\left(\frac{7}{44}\right)\)
\(\chi_{712}(493,\cdot)\) \(-1\) \(1\) \(e\left(\frac{21}{88}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{73}{88}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{43}{88}\right)\) \(e\left(\frac{39}{88}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{31}{88}\right)\) \(e\left(\frac{3}{44}\right)\)
\(\chi_{712}(501,\cdot)\) \(-1\) \(1\) \(e\left(\frac{85}{88}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{65}{88}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{19}{88}\right)\) \(e\left(\frac{7}{88}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{71}{88}\right)\) \(e\left(\frac{31}{44}\right)\)
\(\chi_{712}(541,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{88}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{49}{88}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{59}{88}\right)\) \(e\left(\frac{31}{88}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{63}{88}\right)\) \(e\left(\frac{43}{44}\right)\)
\(\chi_{712}(549,\cdot)\) \(-1\) \(1\) \(e\left(\frac{27}{88}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{31}{88}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{5}{88}\right)\) \(e\left(\frac{25}{88}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{65}{88}\right)\) \(e\left(\frac{29}{44}\right)\)
\(\chi_{712}(557,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{88}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{41}{88}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{35}{88}\right)\) \(e\left(\frac{87}{88}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{15}{88}\right)\) \(e\left(\frac{27}{44}\right)\)
\(\chi_{712}(565,\cdot)\) \(-1\) \(1\) \(e\left(\frac{75}{88}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{47}{88}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{53}{88}\right)\) \(e\left(\frac{1}{88}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{73}{88}\right)\) \(e\left(\frac{17}{44}\right)\)
\(\chi_{712}(597,\cdot)\) \(-1\) \(1\) \(e\left(\frac{39}{88}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{35}{88}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{17}{88}\right)\) \(e\left(\frac{85}{88}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{45}{88}\right)\) \(e\left(\frac{37}{44}\right)\)