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Properties

Label 578.g
Modulus $578$
Conductor $289$
Order $34$
Real no
Primitive no
Minimal yes
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(578, base_ring=CyclotomicField(34)) M = H._module chi = DirichletCharacter(H, M([3])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(33, 578)) 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("578.33"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(578\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(289\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(34\)
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 289.g
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: \(\Q(\zeta_{17})\)
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 34 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(5\) \(7\) \(9\) \(11\) \(13\) \(15\) \(19\) \(21\) \(23\)
\(\chi_{578}(33,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{34}\right)\) \(e\left(\frac{7}{34}\right)\) \(e\left(\frac{23}{34}\right)\) \(e\left(\frac{3}{17}\right)\) \(e\left(\frac{1}{34}\right)\) \(e\left(\frac{5}{17}\right)\) \(e\left(\frac{5}{17}\right)\) \(e\left(\frac{4}{17}\right)\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{23}{34}\right)\)
\(\chi_{578}(67,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{34}\right)\) \(e\left(\frac{31}{34}\right)\) \(e\left(\frac{29}{34}\right)\) \(e\left(\frac{6}{17}\right)\) \(e\left(\frac{19}{34}\right)\) \(e\left(\frac{10}{17}\right)\) \(e\left(\frac{10}{17}\right)\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{9}{17}\right)\) \(e\left(\frac{29}{34}\right)\)
\(\chi_{578}(101,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{34}\right)\) \(e\left(\frac{21}{34}\right)\) \(e\left(\frac{1}{34}\right)\) \(e\left(\frac{9}{17}\right)\) \(e\left(\frac{3}{34}\right)\) \(e\left(\frac{15}{17}\right)\) \(e\left(\frac{15}{17}\right)\) \(e\left(\frac{12}{17}\right)\) \(e\left(\frac{5}{17}\right)\) \(e\left(\frac{1}{34}\right)\)
\(\chi_{578}(135,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{34}\right)\) \(e\left(\frac{11}{34}\right)\) \(e\left(\frac{7}{34}\right)\) \(e\left(\frac{12}{17}\right)\) \(e\left(\frac{21}{34}\right)\) \(e\left(\frac{3}{17}\right)\) \(e\left(\frac{3}{17}\right)\) \(e\left(\frac{16}{17}\right)\) \(e\left(\frac{1}{17}\right)\) \(e\left(\frac{7}{34}\right)\)
\(\chi_{578}(169,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{34}\right)\) \(e\left(\frac{1}{34}\right)\) \(e\left(\frac{13}{34}\right)\) \(e\left(\frac{15}{17}\right)\) \(e\left(\frac{5}{34}\right)\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{3}{17}\right)\) \(e\left(\frac{14}{17}\right)\) \(e\left(\frac{13}{34}\right)\)
\(\chi_{578}(203,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{34}\right)\) \(e\left(\frac{25}{34}\right)\) \(e\left(\frac{19}{34}\right)\) \(e\left(\frac{1}{17}\right)\) \(e\left(\frac{23}{34}\right)\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{7}{17}\right)\) \(e\left(\frac{10}{17}\right)\) \(e\left(\frac{19}{34}\right)\)
\(\chi_{578}(237,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{34}\right)\) \(e\left(\frac{15}{34}\right)\) \(e\left(\frac{25}{34}\right)\) \(e\left(\frac{4}{17}\right)\) \(e\left(\frac{7}{34}\right)\) \(e\left(\frac{1}{17}\right)\) \(e\left(\frac{1}{17}\right)\) \(e\left(\frac{11}{17}\right)\) \(e\left(\frac{6}{17}\right)\) \(e\left(\frac{25}{34}\right)\)
\(\chi_{578}(271,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{34}\right)\) \(e\left(\frac{5}{34}\right)\) \(e\left(\frac{31}{34}\right)\) \(e\left(\frac{7}{17}\right)\) \(e\left(\frac{25}{34}\right)\) \(e\left(\frac{6}{17}\right)\) \(e\left(\frac{6}{17}\right)\) \(e\left(\frac{15}{17}\right)\) \(e\left(\frac{2}{17}\right)\) \(e\left(\frac{31}{34}\right)\)
\(\chi_{578}(305,\cdot)\) \(1\) \(1\) \(e\left(\frac{27}{34}\right)\) \(e\left(\frac{29}{34}\right)\) \(e\left(\frac{3}{34}\right)\) \(e\left(\frac{10}{17}\right)\) \(e\left(\frac{9}{34}\right)\) \(e\left(\frac{11}{17}\right)\) \(e\left(\frac{11}{17}\right)\) \(e\left(\frac{2}{17}\right)\) \(e\left(\frac{15}{17}\right)\) \(e\left(\frac{3}{34}\right)\)
\(\chi_{578}(339,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{34}\right)\) \(e\left(\frac{19}{34}\right)\) \(e\left(\frac{9}{34}\right)\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{27}{34}\right)\) \(e\left(\frac{16}{17}\right)\) \(e\left(\frac{16}{17}\right)\) \(e\left(\frac{6}{17}\right)\) \(e\left(\frac{11}{17}\right)\) \(e\left(\frac{9}{34}\right)\)
\(\chi_{578}(373,\cdot)\) \(1\) \(1\) \(e\left(\frac{33}{34}\right)\) \(e\left(\frac{9}{34}\right)\) \(e\left(\frac{15}{34}\right)\) \(e\left(\frac{16}{17}\right)\) \(e\left(\frac{11}{34}\right)\) \(e\left(\frac{4}{17}\right)\) \(e\left(\frac{4}{17}\right)\) \(e\left(\frac{10}{17}\right)\) \(e\left(\frac{7}{17}\right)\) \(e\left(\frac{15}{34}\right)\)
\(\chi_{578}(407,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{34}\right)\) \(e\left(\frac{33}{34}\right)\) \(e\left(\frac{21}{34}\right)\) \(e\left(\frac{2}{17}\right)\) \(e\left(\frac{29}{34}\right)\) \(e\left(\frac{9}{17}\right)\) \(e\left(\frac{9}{17}\right)\) \(e\left(\frac{14}{17}\right)\) \(e\left(\frac{3}{17}\right)\) \(e\left(\frac{21}{34}\right)\)
\(\chi_{578}(441,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{34}\right)\) \(e\left(\frac{23}{34}\right)\) \(e\left(\frac{27}{34}\right)\) \(e\left(\frac{5}{17}\right)\) \(e\left(\frac{13}{34}\right)\) \(e\left(\frac{14}{17}\right)\) \(e\left(\frac{14}{17}\right)\) \(e\left(\frac{1}{17}\right)\) \(e\left(\frac{16}{17}\right)\) \(e\left(\frac{27}{34}\right)\)
\(\chi_{578}(475,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{34}\right)\) \(e\left(\frac{13}{34}\right)\) \(e\left(\frac{33}{34}\right)\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{31}{34}\right)\) \(e\left(\frac{2}{17}\right)\) \(e\left(\frac{2}{17}\right)\) \(e\left(\frac{5}{17}\right)\) \(e\left(\frac{12}{17}\right)\) \(e\left(\frac{33}{34}\right)\)
\(\chi_{578}(509,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{34}\right)\) \(e\left(\frac{3}{34}\right)\) \(e\left(\frac{5}{34}\right)\) \(e\left(\frac{11}{17}\right)\) \(e\left(\frac{15}{34}\right)\) \(e\left(\frac{7}{17}\right)\) \(e\left(\frac{7}{17}\right)\) \(e\left(\frac{9}{17}\right)\) \(e\left(\frac{8}{17}\right)\) \(e\left(\frac{5}{34}\right)\)
\(\chi_{578}(543,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{34}\right)\) \(e\left(\frac{27}{34}\right)\) \(e\left(\frac{11}{34}\right)\) \(e\left(\frac{14}{17}\right)\) \(e\left(\frac{33}{34}\right)\) \(e\left(\frac{12}{17}\right)\) \(e\left(\frac{12}{17}\right)\) \(e\left(\frac{13}{17}\right)\) \(e\left(\frac{4}{17}\right)\) \(e\left(\frac{11}{34}\right)\)