Properties

Label 405.x
Modulus $405$
Conductor $405$
Order $108$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(405, base_ring=CyclotomicField(108))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,27]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(2,405))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(405\)
Conductor: \(405\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(108\)
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: $\Q(\zeta_{108})$
Fixed field: Number field defined by a degree 108 polynomial (not computed)

First 31 of 36 characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(7\) \(8\) \(11\) \(13\) \(14\) \(16\) \(17\) \(19\)
\(\chi_{405}(2,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{108}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{59}{108}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{97}{108}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{405}(23,\cdot)\) \(1\) \(1\) \(e\left(\frac{103}{108}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{1}{108}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{95}{108}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{405}(32,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{108}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{79}{108}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{53}{108}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{405}(38,\cdot)\) \(1\) \(1\) \(e\left(\frac{71}{108}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{29}{108}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{55}{108}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{1}{18}\right)\)
\(\chi_{405}(47,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{108}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{35}{108}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{85}{108}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{405}(68,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{108}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{13}{108}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{47}{108}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{405}(77,\cdot)\) \(1\) \(1\) \(e\left(\frac{85}{108}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{91}{108}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{5}{108}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{405}(83,\cdot)\) \(1\) \(1\) \(e\left(\frac{83}{108}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{5}{108}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{43}{108}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{405}(92,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{108}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{11}{108}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{73}{108}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{1}{18}\right)\)
\(\chi_{405}(113,\cdot)\) \(1\) \(1\) \(e\left(\frac{91}{108}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{25}{108}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{107}{108}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{405}(122,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{108}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{103}{108}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{65}{108}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{405}(128,\cdot)\) \(1\) \(1\) \(e\left(\frac{95}{108}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{89}{108}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{31}{108}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{405}(137,\cdot)\) \(1\) \(1\) \(e\left(\frac{65}{108}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{95}{108}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{61}{108}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{405}(158,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{108}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{37}{108}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{59}{108}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{405}(167,\cdot)\) \(1\) \(1\) \(e\left(\frac{73}{108}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{7}{108}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{17}{108}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{405}(173,\cdot)\) \(1\) \(1\) \(e\left(\frac{107}{108}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{65}{108}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{19}{108}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{1}{18}\right)\)
\(\chi_{405}(182,\cdot)\) \(1\) \(1\) \(e\left(\frac{77}{108}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{71}{108}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{49}{108}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{405}(203,\cdot)\) \(1\) \(1\) \(e\left(\frac{79}{108}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{49}{108}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{11}{108}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{405}(212,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{108}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{19}{108}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{77}{108}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{405}(218,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{108}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{41}{108}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{7}{108}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{405}(227,\cdot)\) \(1\) \(1\) \(e\left(\frac{89}{108}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{47}{108}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{37}{108}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{1}{18}\right)\)
\(\chi_{405}(248,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{108}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{61}{108}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{71}{108}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{405}(257,\cdot)\) \(1\) \(1\) \(e\left(\frac{61}{108}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{31}{108}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{29}{108}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{405}(263,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{108}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{17}{108}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{103}{108}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{405}(272,\cdot)\) \(1\) \(1\) \(e\left(\frac{101}{108}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{23}{108}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{25}{108}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{7}{18}\right)\)
\(\chi_{405}(293,\cdot)\) \(1\) \(1\) \(e\left(\frac{67}{108}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{73}{108}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{23}{108}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{5}{18}\right)\)
\(\chi_{405}(302,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{108}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{43}{108}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{89}{108}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{405}(308,\cdot)\) \(1\) \(1\) \(e\left(\frac{35}{108}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{101}{108}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{91}{108}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{1}{18}\right)\)
\(\chi_{405}(317,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{108}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{107}{108}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{13}{108}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{13}{18}\right)\)
\(\chi_{405}(338,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{108}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{85}{108}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{83}{108}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{11}{18}\right)\)
\(\chi_{405}(347,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{108}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{55}{108}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{41}{108}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{5}{18}\right)\)