Properties

Label 675.bj
Modulus $675$
Conductor $675$
Order $180$
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(675, base_ring=CyclotomicField(180))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([80,171]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(13,675))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(675\)
Conductor: \(675\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

First 31 of 48 characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(7\) \(8\) \(11\) \(13\) \(14\) \(16\) \(17\) \(19\)
\(\chi_{675}(13,\cdot)\) \(-1\) \(1\) \(e\left(\frac{71}{180}\right)\) \(e\left(\frac{71}{90}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{109}{180}\right)\) \(e\left(\frac{23}{90}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{13}{30}\right)\)
\(\chi_{675}(22,\cdot)\) \(-1\) \(1\) \(e\left(\frac{113}{180}\right)\) \(e\left(\frac{23}{90}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{67}{180}\right)\) \(e\left(\frac{29}{90}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{19}{30}\right)\)
\(\chi_{675}(52,\cdot)\) \(-1\) \(1\) \(e\left(\frac{109}{180}\right)\) \(e\left(\frac{19}{90}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{71}{180}\right)\) \(e\left(\frac{67}{90}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{17}{30}\right)\)
\(\chi_{675}(58,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{180}\right)\) \(e\left(\frac{47}{90}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{133}{180}\right)\) \(e\left(\frac{71}{90}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{1}{30}\right)\)
\(\chi_{675}(67,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{180}\right)\) \(e\left(\frac{17}{90}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{163}{180}\right)\) \(e\left(\frac{41}{90}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{1}{30}\right)\)
\(\chi_{675}(88,\cdot)\) \(-1\) \(1\) \(e\left(\frac{151}{180}\right)\) \(e\left(\frac{61}{90}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{29}{180}\right)\) \(e\left(\frac{73}{90}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{23}{30}\right)\)
\(\chi_{675}(97,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{180}\right)\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{167}{180}\right)\) \(e\left(\frac{79}{90}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{29}{30}\right)\)
\(\chi_{675}(103,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{180}\right)\) \(e\left(\frac{23}{90}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{157}{180}\right)\) \(e\left(\frac{29}{90}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{19}{30}\right)\)
\(\chi_{675}(112,\cdot)\) \(-1\) \(1\) \(e\left(\frac{101}{180}\right)\) \(e\left(\frac{11}{90}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{79}{180}\right)\) \(e\left(\frac{53}{90}\right)\) \(e\left(\frac{11}{45}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{13}{30}\right)\)
\(\chi_{675}(133,\cdot)\) \(-1\) \(1\) \(e\left(\frac{127}{180}\right)\) \(e\left(\frac{37}{90}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{53}{180}\right)\) \(e\left(\frac{31}{90}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{11}{30}\right)\)
\(\chi_{675}(142,\cdot)\) \(-1\) \(1\) \(e\left(\frac{97}{180}\right)\) \(e\left(\frac{7}{90}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{83}{180}\right)\) \(e\left(\frac{1}{90}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{11}{30}\right)\)
\(\chi_{675}(148,\cdot)\) \(-1\) \(1\) \(e\left(\frac{179}{180}\right)\) \(e\left(\frac{89}{90}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{1}{180}\right)\) \(e\left(\frac{77}{90}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{7}{30}\right)\)
\(\chi_{675}(178,\cdot)\) \(-1\) \(1\) \(e\left(\frac{103}{180}\right)\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{77}{180}\right)\) \(e\left(\frac{79}{90}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{29}{30}\right)\)
\(\chi_{675}(187,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{180}\right)\) \(e\left(\frac{1}{90}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{179}{180}\right)\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{23}{30}\right)\)
\(\chi_{675}(202,\cdot)\) \(-1\) \(1\) \(e\left(\frac{89}{180}\right)\) \(e\left(\frac{89}{90}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{91}{180}\right)\) \(e\left(\frac{77}{90}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{7}{30}\right)\)
\(\chi_{675}(223,\cdot)\) \(-1\) \(1\) \(e\left(\frac{79}{180}\right)\) \(e\left(\frac{79}{90}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{101}{180}\right)\) \(e\left(\frac{37}{90}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{17}{30}\right)\)
\(\chi_{675}(238,\cdot)\) \(-1\) \(1\) \(e\left(\frac{131}{180}\right)\) \(e\left(\frac{41}{90}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{49}{180}\right)\) \(e\left(\frac{83}{90}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{13}{30}\right)\)
\(\chi_{675}(247,\cdot)\) \(-1\) \(1\) \(e\left(\frac{173}{180}\right)\) \(e\left(\frac{83}{90}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{7}{180}\right)\) \(e\left(\frac{89}{90}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{19}{30}\right)\)
\(\chi_{675}(277,\cdot)\) \(-1\) \(1\) \(e\left(\frac{169}{180}\right)\) \(e\left(\frac{79}{90}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{11}{180}\right)\) \(e\left(\frac{37}{90}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{17}{30}\right)\)
\(\chi_{675}(283,\cdot)\) \(-1\) \(1\) \(e\left(\frac{107}{180}\right)\) \(e\left(\frac{17}{90}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{73}{180}\right)\) \(e\left(\frac{41}{90}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{1}{30}\right)\)
\(\chi_{675}(292,\cdot)\) \(-1\) \(1\) \(e\left(\frac{77}{180}\right)\) \(e\left(\frac{77}{90}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{23}{45}\right)\) \(e\left(\frac{103}{180}\right)\) \(e\left(\frac{11}{90}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{1}{30}\right)\)
\(\chi_{675}(313,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{180}\right)\) \(e\left(\frac{31}{90}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{149}{180}\right)\) \(e\left(\frac{43}{90}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{23}{30}\right)\)
\(\chi_{675}(322,\cdot)\) \(-1\) \(1\) \(e\left(\frac{73}{180}\right)\) \(e\left(\frac{73}{90}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{107}{180}\right)\) \(e\left(\frac{49}{90}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{29}{30}\right)\)
\(\chi_{675}(328,\cdot)\) \(-1\) \(1\) \(e\left(\frac{83}{180}\right)\) \(e\left(\frac{83}{90}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{97}{180}\right)\) \(e\left(\frac{89}{90}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{19}{30}\right)\)
\(\chi_{675}(337,\cdot)\) \(-1\) \(1\) \(e\left(\frac{161}{180}\right)\) \(e\left(\frac{71}{90}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{19}{180}\right)\) \(e\left(\frac{23}{90}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{13}{30}\right)\)
\(\chi_{675}(358,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{180}\right)\) \(e\left(\frac{7}{90}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{173}{180}\right)\) \(e\left(\frac{1}{90}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{11}{30}\right)\)
\(\chi_{675}(367,\cdot)\) \(-1\) \(1\) \(e\left(\frac{157}{180}\right)\) \(e\left(\frac{67}{90}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{23}{180}\right)\) \(e\left(\frac{61}{90}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{11}{30}\right)\)
\(\chi_{675}(373,\cdot)\) \(-1\) \(1\) \(e\left(\frac{59}{180}\right)\) \(e\left(\frac{59}{90}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{121}{180}\right)\) \(e\left(\frac{47}{90}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{7}{30}\right)\)
\(\chi_{675}(403,\cdot)\) \(-1\) \(1\) \(e\left(\frac{163}{180}\right)\) \(e\left(\frac{73}{90}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{17}{180}\right)\) \(e\left(\frac{49}{90}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{29}{30}\right)\)
\(\chi_{675}(412,\cdot)\) \(-1\) \(1\) \(e\left(\frac{61}{180}\right)\) \(e\left(\frac{61}{90}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{119}{180}\right)\) \(e\left(\frac{73}{90}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{23}{30}\right)\)
\(\chi_{675}(427,\cdot)\) \(-1\) \(1\) \(e\left(\frac{149}{180}\right)\) \(e\left(\frac{59}{90}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{31}{180}\right)\) \(e\left(\frac{47}{90}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{7}{30}\right)\)