Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character orbit
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(44)) M = H._module chi = DirichletCharacter(H, M([11,38])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(7, 230)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("230.7"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(230\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(115\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(44\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: no, induced from 115.l
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Related number fields

Field of values: \(\Q(\zeta_{44})\)
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: \(\Q(\zeta_{115})^+\)
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(7\) \(9\) \(11\) \(13\) \(17\) \(19\) \(21\) \(27\) \(29\)
\(\chi_{230}(7,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{1}{22}\right)\)
\(\chi_{230}(17,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{5}{22}\right)\)
\(\chi_{230}(33,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{21}{22}\right)\)
\(\chi_{230}(37,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{15}{22}\right)\)
\(\chi_{230}(43,\cdot)\) \(1\) \(1\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{13}{22}\right)\)
\(\chi_{230}(53,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{1}{22}\right)\)
\(\chi_{230}(57,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{19}{22}\right)\)
\(\chi_{230}(63,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{5}{22}\right)\)
\(\chi_{230}(67,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{3}{22}\right)\)
\(\chi_{230}(83,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{15}{22}\right)\)
\(\chi_{230}(97,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{7}{22}\right)\)
\(\chi_{230}(103,\cdot)\) \(1\) \(1\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{19}{22}\right)\)
\(\chi_{230}(107,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{9}{22}\right)\)
\(\chi_{230}(113,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{3}{22}\right)\)
\(\chi_{230}(143,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{7}{22}\right)\)
\(\chi_{230}(153,\cdot)\) \(1\) \(1\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{9}{22}\right)\)
\(\chi_{230}(157,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{17}{22}\right)\)
\(\chi_{230}(203,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{17}{22}\right)\)
\(\chi_{230}(217,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{35}{44}\right)\) \(e\left(\frac{21}{22}\right)\)
\(\chi_{230}(227,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{13}{22}\right)\)