Properties

Label 605.r
Modulus $605$
Conductor $605$
Order $44$
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(605, base_ring=CyclotomicField(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,2]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(32,605))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(605\)
Conductor: \(605\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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: \(\Q(\zeta_{44})\)
Fixed field: 44.44.2885428559557085084648615903962269104974580506944665166312236845353556846511909399754484184086322784423828125.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(12\) \(13\) \(14\)
\(\chi_{605}(32,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{44}\right)\) \(-i\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{39}{44}\right)\) \(-1\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{19}{22}\right)\)
\(\chi_{605}(43,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{44}\right)\) \(i\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{41}{44}\right)\) \(-1\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{7}{22}\right)\)
\(\chi_{605}(87,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{44}\right)\) \(-i\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{27}{44}\right)\) \(-1\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{3}{22}\right)\)
\(\chi_{605}(98,\cdot)\) \(1\) \(1\) \(e\left(\frac{39}{44}\right)\) \(i\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{29}{44}\right)\) \(-1\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{13}{22}\right)\)
\(\chi_{605}(142,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{44}\right)\) \(-i\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{15}{44}\right)\) \(-1\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{9}{22}\right)\)
\(\chi_{605}(153,\cdot)\) \(1\) \(1\) \(e\left(\frac{35}{44}\right)\) \(i\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{17}{44}\right)\) \(-1\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{19}{22}\right)\)
\(\chi_{605}(197,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{44}\right)\) \(-i\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{3}{44}\right)\) \(-1\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{15}{22}\right)\)
\(\chi_{605}(208,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{44}\right)\) \(i\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{5}{44}\right)\) \(-1\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{3}{22}\right)\)
\(\chi_{605}(252,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{44}\right)\) \(-i\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{35}{44}\right)\) \(-1\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{21}{22}\right)\)
\(\chi_{605}(263,\cdot)\) \(1\) \(1\) \(e\left(\frac{27}{44}\right)\) \(i\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{37}{44}\right)\) \(-1\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{9}{22}\right)\)
\(\chi_{605}(307,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{44}\right)\) \(-i\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{23}{44}\right)\) \(-1\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{5}{22}\right)\)
\(\chi_{605}(318,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{44}\right)\) \(i\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{25}{44}\right)\) \(-1\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{15}{22}\right)\)
\(\chi_{605}(373,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{44}\right)\) \(i\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{13}{44}\right)\) \(-1\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{21}{22}\right)\)
\(\chi_{605}(417,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{44}\right)\) \(-i\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{43}{44}\right)\) \(-1\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{17}{22}\right)\)
\(\chi_{605}(428,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{44}\right)\) \(i\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{1}{44}\right)\) \(-1\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{5}{22}\right)\)
\(\chi_{605}(472,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{44}\right)\) \(-i\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{31}{44}\right)\) \(-1\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{1}{22}\right)\)
\(\chi_{605}(527,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{44}\right)\) \(-i\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{19}{44}\right)\) \(-1\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{7}{22}\right)\)
\(\chi_{605}(538,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{44}\right)\) \(i\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{21}{44}\right)\) \(-1\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{17}{22}\right)\)
\(\chi_{605}(582,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{44}\right)\) \(-i\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{7}{44}\right)\) \(-1\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{13}{22}\right)\)
\(\chi_{605}(593,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{44}\right)\) \(i\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{9}{44}\right)\) \(-1\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{1}{22}\right)\)