Properties

Label 648.bd
Modulus $648$
Conductor $648$
Order $54$
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(648, base_ring=CyclotomicField(54))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,27,8]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(13,648))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(648\)
Conductor: \(648\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 54 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\)
\(\chi_{648}(13,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{26}{27}\right)\)
\(\chi_{648}(61,\cdot)\) \(1\) \(1\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{7}{27}\right)\)
\(\chi_{648}(85,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{20}{27}\right)\)
\(\chi_{648}(133,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{19}{27}\right)\)
\(\chi_{648}(157,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{14}{27}\right)\)
\(\chi_{648}(205,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{4}{27}\right)\)
\(\chi_{648}(229,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{8}{27}\right)\)
\(\chi_{648}(277,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{16}{27}\right)\)
\(\chi_{648}(301,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{2}{27}\right)\)
\(\chi_{648}(349,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{1}{27}\right)\)
\(\chi_{648}(373,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{23}{27}\right)\)
\(\chi_{648}(421,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{13}{27}\right)\)
\(\chi_{648}(445,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{17}{27}\right)\)
\(\chi_{648}(493,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{25}{27}\right)\)
\(\chi_{648}(517,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{11}{27}\right)\)
\(\chi_{648}(565,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{10}{27}\right)\)
\(\chi_{648}(589,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{5}{27}\right)\)
\(\chi_{648}(637,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{22}{27}\right)\)