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Properties

Label 621.s
Modulus $621$
Conductor $207$
Order $66$
Real no
Primitive no
Minimal no
Parity even

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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(621, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([55,21])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(17, 621)) 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("621.17"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(621\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(207\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(66\)
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 207.o
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: no
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: \(\Q(\zeta_{33})\)
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: Number field defined by a degree 66 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(7\) \(8\) \(10\) \(11\) \(13\) \(14\) \(16\)
\(\chi_{621}(17,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{66}\right)\) \(e\left(\frac{31}{33}\right)\) \(e\left(\frac{16}{33}\right)\) \(e\left(\frac{25}{66}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{4}{33}\right)\) \(e\left(\frac{28}{33}\right)\) \(e\left(\frac{29}{33}\right)\)
\(\chi_{621}(44,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{66}\right)\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{37}{66}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{5}{33}\right)\) \(e\left(\frac{31}{33}\right)\) \(e\left(\frac{19}{33}\right)\) \(e\left(\frac{2}{33}\right)\)
\(\chi_{621}(89,\cdot)\) \(1\) \(1\) \(e\left(\frac{41}{66}\right)\) \(e\left(\frac{8}{33}\right)\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{65}{66}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{17}{33}\right)\) \(e\left(\frac{20}{33}\right)\) \(e\left(\frac{16}{33}\right)\)
\(\chi_{621}(125,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{66}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{61}{66}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{19}{33}\right)\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{14}{33}\right)\)
\(\chi_{621}(143,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{66}\right)\) \(e\left(\frac{17}{33}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{35}{66}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{19}{33}\right)\) \(e\left(\frac{32}{33}\right)\) \(e\left(\frac{26}{33}\right)\) \(e\left(\frac{1}{33}\right)\)
\(\chi_{621}(152,\cdot)\) \(1\) \(1\) \(e\left(\frac{49}{66}\right)\) \(e\left(\frac{16}{33}\right)\) \(e\left(\frac{4}{33}\right)\) \(e\left(\frac{31}{66}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{32}{33}\right)\)
\(\chi_{621}(224,\cdot)\) \(1\) \(1\) \(e\left(\frac{53}{66}\right)\) \(e\left(\frac{20}{33}\right)\) \(e\left(\frac{5}{33}\right)\) \(e\left(\frac{47}{66}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{26}{33}\right)\) \(e\left(\frac{17}{33}\right)\) \(e\left(\frac{7}{33}\right)\)
\(\chi_{621}(251,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{66}\right)\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{59}{66}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{16}{33}\right)\) \(e\left(\frac{20}{33}\right)\) \(e\left(\frac{8}{33}\right)\) \(e\left(\frac{13}{33}\right)\)
\(\chi_{621}(260,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{66}\right)\) \(e\left(\frac{4}{33}\right)\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{49}{66}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{20}{33}\right)\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{8}{33}\right)\)
\(\chi_{621}(287,\cdot)\) \(1\) \(1\) \(e\left(\frac{43}{66}\right)\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{19}{33}\right)\) \(e\left(\frac{7}{66}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{17}{33}\right)\) \(e\left(\frac{13}{33}\right)\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{20}{33}\right)\)
\(\chi_{621}(314,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{66}\right)\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{31}{33}\right)\) \(e\left(\frac{1}{66}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{26}{33}\right)\) \(e\left(\frac{16}{33}\right)\) \(e\left(\frac{13}{33}\right)\) \(e\left(\frac{17}{33}\right)\)
\(\chi_{621}(332,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{66}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{32}{33}\right)\) \(e\left(\frac{17}{66}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{13}{33}\right)\) \(e\left(\frac{8}{33}\right)\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{25}{33}\right)\)
\(\chi_{621}(341,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{66}\right)\) \(e\left(\frac{13}{33}\right)\) \(e\left(\frac{28}{33}\right)\) \(e\left(\frac{19}{66}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{32}{33}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{16}{33}\right)\) \(e\left(\frac{26}{33}\right)\)
\(\chi_{621}(359,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{66}\right)\) \(e\left(\frac{5}{33}\right)\) \(e\left(\frac{26}{33}\right)\) \(e\left(\frac{53}{66}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{10}{33}\right)\)
\(\chi_{621}(467,\cdot)\) \(1\) \(1\) \(e\left(\frac{59}{66}\right)\) \(e\left(\frac{26}{33}\right)\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{5}{66}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{31}{33}\right)\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{32}{33}\right)\) \(e\left(\frac{19}{33}\right)\)
\(\chi_{621}(494,\cdot)\) \(1\) \(1\) \(e\left(\frac{65}{66}\right)\) \(e\left(\frac{32}{33}\right)\) \(e\left(\frac{8}{33}\right)\) \(e\left(\frac{29}{66}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{28}{33}\right)\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{31}{33}\right)\)
\(\chi_{621}(503,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{66}\right)\) \(e\left(\frac{19}{33}\right)\) \(e\left(\frac{13}{33}\right)\) \(e\left(\frac{43}{66}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{28}{33}\right)\) \(e\left(\frac{31}{33}\right)\) \(e\left(\frac{5}{33}\right)\)
\(\chi_{621}(521,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{66}\right)\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{20}{33}\right)\) \(e\left(\frac{23}{66}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{4}{33}\right)\) \(e\left(\frac{5}{33}\right)\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{28}{33}\right)\)
\(\chi_{621}(548,\cdot)\) \(1\) \(1\) \(e\left(\frac{35}{66}\right)\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{17}{33}\right)\) \(e\left(\frac{41}{66}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{5}{33}\right)\) \(e\left(\frac{4}{33}\right)\)
\(\chi_{621}(557,\cdot)\) \(1\) \(1\) \(e\left(\frac{61}{66}\right)\) \(e\left(\frac{28}{33}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{13}{66}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{8}{33}\right)\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{4}{33}\right)\) \(e\left(\frac{23}{33}\right)\)