Properties

Label 1045.cs
Modulus $1045$
Conductor $1045$
Order $180$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(180))
 
M = H._module
 
chi = DirichletCharacter(H, M([45,162,100]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(17,1045))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1045\)
Conductor: \(1045\)
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: even
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 \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(12\) \(13\) \(14\)
\(\chi_{1045}(17,\cdot)\) \(1\) \(1\) \(e\left(\frac{127}{180}\right)\) \(e\left(\frac{31}{180}\right)\) \(e\left(\frac{37}{90}\right)\) \(e\left(\frac{79}{90}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{31}{90}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{77}{180}\right)\) \(e\left(\frac{53}{90}\right)\)
\(\chi_{1045}(28,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{180}\right)\) \(e\left(\frac{41}{180}\right)\) \(e\left(\frac{17}{90}\right)\) \(e\left(\frac{29}{90}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{41}{90}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{67}{180}\right)\) \(e\left(\frac{73}{90}\right)\)
\(\chi_{1045}(62,\cdot)\) \(1\) \(1\) \(e\left(\frac{151}{180}\right)\) \(e\left(\frac{163}{180}\right)\) \(e\left(\frac{61}{90}\right)\) \(e\left(\frac{67}{90}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{73}{90}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{161}{180}\right)\) \(e\left(\frac{29}{90}\right)\)
\(\chi_{1045}(63,\cdot)\) \(1\) \(1\) \(e\left(\frac{149}{180}\right)\) \(e\left(\frac{137}{180}\right)\) \(e\left(\frac{59}{90}\right)\) \(e\left(\frac{53}{90}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{47}{90}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{79}{180}\right)\) \(e\left(\frac{31}{90}\right)\)
\(\chi_{1045}(73,\cdot)\) \(1\) \(1\) \(e\left(\frac{121}{180}\right)\) \(e\left(\frac{133}{180}\right)\) \(e\left(\frac{31}{90}\right)\) \(e\left(\frac{37}{90}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{43}{90}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{180}\right)\) \(e\left(\frac{59}{90}\right)\)
\(\chi_{1045}(112,\cdot)\) \(1\) \(1\) \(e\left(\frac{163}{180}\right)\) \(e\left(\frac{139}{180}\right)\) \(e\left(\frac{73}{90}\right)\) \(e\left(\frac{61}{90}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{49}{90}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{113}{180}\right)\) \(e\left(\frac{17}{90}\right)\)
\(\chi_{1045}(118,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{180}\right)\) \(e\left(\frac{17}{180}\right)\) \(e\left(\frac{29}{90}\right)\) \(e\left(\frac{23}{90}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{17}{90}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{19}{180}\right)\) \(e\left(\frac{61}{90}\right)\)
\(\chi_{1045}(123,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{180}\right)\) \(e\left(\frac{149}{180}\right)\) \(e\left(\frac{53}{90}\right)\) \(e\left(\frac{11}{90}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{59}{90}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{103}{180}\right)\) \(e\left(\frac{37}{90}\right)\)
\(\chi_{1045}(138,\cdot)\) \(1\) \(1\) \(e\left(\frac{97}{180}\right)\) \(e\left(\frac{1}{180}\right)\) \(e\left(\frac{7}{90}\right)\) \(e\left(\frac{49}{90}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{1}{90}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{107}{180}\right)\) \(e\left(\frac{83}{90}\right)\)
\(\chi_{1045}(233,\cdot)\) \(1\) \(1\) \(e\left(\frac{133}{180}\right)\) \(e\left(\frac{109}{180}\right)\) \(e\left(\frac{43}{90}\right)\) \(e\left(\frac{31}{90}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{19}{90}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{143}{180}\right)\) \(e\left(\frac{47}{90}\right)\)
\(\chi_{1045}(237,\cdot)\) \(1\) \(1\) \(e\left(\frac{107}{180}\right)\) \(e\left(\frac{131}{180}\right)\) \(e\left(\frac{17}{90}\right)\) \(e\left(\frac{29}{90}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{41}{90}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{157}{180}\right)\) \(e\left(\frac{73}{90}\right)\)
\(\chi_{1045}(272,\cdot)\) \(1\) \(1\) \(e\left(\frac{59}{180}\right)\) \(e\left(\frac{47}{180}\right)\) \(e\left(\frac{59}{90}\right)\) \(e\left(\frac{53}{90}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{47}{90}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{169}{180}\right)\) \(e\left(\frac{31}{90}\right)\)
\(\chi_{1045}(282,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{180}\right)\) \(e\left(\frac{43}{180}\right)\) \(e\left(\frac{31}{90}\right)\) \(e\left(\frac{37}{90}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{43}{90}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{101}{180}\right)\) \(e\left(\frac{59}{90}\right)\)
\(\chi_{1045}(283,\cdot)\) \(1\) \(1\) \(e\left(\frac{109}{180}\right)\) \(e\left(\frac{157}{180}\right)\) \(e\left(\frac{19}{90}\right)\) \(e\left(\frac{43}{90}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{67}{90}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{59}{180}\right)\) \(e\left(\frac{71}{90}\right)\)
\(\chi_{1045}(327,\cdot)\) \(1\) \(1\) \(e\left(\frac{119}{180}\right)\) \(e\left(\frac{107}{180}\right)\) \(e\left(\frac{29}{90}\right)\) \(e\left(\frac{23}{90}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{17}{90}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{109}{180}\right)\) \(e\left(\frac{61}{90}\right)\)
\(\chi_{1045}(332,\cdot)\) \(1\) \(1\) \(e\left(\frac{143}{180}\right)\) \(e\left(\frac{59}{180}\right)\) \(e\left(\frac{53}{90}\right)\) \(e\left(\frac{11}{90}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{59}{90}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{13}{180}\right)\) \(e\left(\frac{37}{90}\right)\)
\(\chi_{1045}(347,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{180}\right)\) \(e\left(\frac{91}{180}\right)\) \(e\left(\frac{7}{90}\right)\) \(e\left(\frac{49}{90}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{1}{90}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{17}{180}\right)\) \(e\left(\frac{83}{90}\right)\)
\(\chi_{1045}(348,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{180}\right)\) \(e\left(\frac{173}{180}\right)\) \(e\left(\frac{41}{90}\right)\) \(e\left(\frac{17}{90}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{83}{90}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{151}{180}\right)\) \(e\left(\frac{49}{90}\right)\)
\(\chi_{1045}(358,\cdot)\) \(1\) \(1\) \(e\left(\frac{157}{180}\right)\) \(e\left(\frac{61}{180}\right)\) \(e\left(\frac{67}{90}\right)\) \(e\left(\frac{19}{90}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{61}{90}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{47}{180}\right)\) \(e\left(\frac{23}{90}\right)\)
\(\chi_{1045}(403,\cdot)\) \(1\) \(1\) \(e\left(\frac{101}{180}\right)\) \(e\left(\frac{53}{180}\right)\) \(e\left(\frac{11}{90}\right)\) \(e\left(\frac{77}{90}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{53}{90}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{91}{180}\right)\) \(e\left(\frac{79}{90}\right)\)
\(\chi_{1045}(442,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{180}\right)\) \(e\left(\frac{19}{180}\right)\) \(e\left(\frac{43}{90}\right)\) \(e\left(\frac{31}{90}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{19}{90}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{53}{180}\right)\) \(e\left(\frac{47}{90}\right)\)
\(\chi_{1045}(453,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{180}\right)\) \(e\left(\frac{169}{180}\right)\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{1}{90}\right)\) \(e\left(\frac{47}{60}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{79}{90}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{83}{180}\right)\) \(e\left(\frac{77}{90}\right)\)
\(\chi_{1045}(492,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{180}\right)\) \(e\left(\frac{67}{180}\right)\) \(e\left(\frac{19}{90}\right)\) \(e\left(\frac{43}{90}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{67}{90}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{149}{180}\right)\) \(e\left(\frac{71}{90}\right)\)
\(\chi_{1045}(503,\cdot)\) \(1\) \(1\) \(e\left(\frac{89}{180}\right)\) \(e\left(\frac{77}{180}\right)\) \(e\left(\frac{89}{90}\right)\) \(e\left(\frac{83}{90}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{77}{90}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{139}{180}\right)\) \(e\left(\frac{1}{90}\right)\)
\(\chi_{1045}(557,\cdot)\) \(1\) \(1\) \(e\left(\frac{131}{180}\right)\) \(e\left(\frac{83}{180}\right)\) \(e\left(\frac{41}{90}\right)\) \(e\left(\frac{17}{90}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{83}{90}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{61}{180}\right)\) \(e\left(\frac{49}{90}\right)\)
\(\chi_{1045}(567,\cdot)\) \(1\) \(1\) \(e\left(\frac{67}{180}\right)\) \(e\left(\frac{151}{180}\right)\) \(e\left(\frac{67}{90}\right)\) \(e\left(\frac{19}{90}\right)\) \(e\left(\frac{53}{60}\right)\) \(e\left(\frac{7}{60}\right)\) \(e\left(\frac{61}{90}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{137}{180}\right)\) \(e\left(\frac{23}{90}\right)\)
\(\chi_{1045}(568,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{180}\right)\) \(e\left(\frac{13}{180}\right)\) \(e\left(\frac{1}{90}\right)\) \(e\left(\frac{7}{90}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{1}{60}\right)\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{131}{180}\right)\) \(e\left(\frac{89}{90}\right)\)
\(\chi_{1045}(612,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{180}\right)\) \(e\left(\frac{143}{180}\right)\) \(e\left(\frac{11}{90}\right)\) \(e\left(\frac{77}{90}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{53}{90}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{180}\right)\) \(e\left(\frac{79}{90}\right)\)
\(\chi_{1045}(613,\cdot)\) \(1\) \(1\) \(e\left(\frac{169}{180}\right)\) \(e\left(\frac{37}{180}\right)\) \(e\left(\frac{79}{90}\right)\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{37}{90}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{179}{180}\right)\) \(e\left(\frac{11}{90}\right)\)
\(\chi_{1045}(633,\cdot)\) \(1\) \(1\) \(e\left(\frac{77}{180}\right)\) \(e\left(\frac{101}{180}\right)\) \(e\left(\frac{77}{90}\right)\) \(e\left(\frac{89}{90}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{11}{90}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{180}\right)\) \(e\left(\frac{13}{90}\right)\)
\(\chi_{1045}(662,\cdot)\) \(1\) \(1\) \(e\left(\frac{103}{180}\right)\) \(e\left(\frac{79}{180}\right)\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{1}{90}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{79}{90}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{173}{180}\right)\) \(e\left(\frac{77}{90}\right)\)