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Properties

Label 2370.be
Modulus $2370$
Conductor $237$
Order $26$
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(2370, base_ring=CyclotomicField(26)) M = H._module chi = DirichletCharacter(H, M([13,0,25])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(41, 2370)) 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("2370.41"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(2370\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(237\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(26\)
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 237.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: \(\Q(\zeta_{13})\)
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: 26.26.439798347913742167434960513305495870460482985171691677.1
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Characters in Galois orbit

Character \(-1\) \(1\) \(7\) \(11\) \(13\) \(17\) \(19\) \(23\) \(29\) \(31\) \(37\) \(41\)
\(\chi_{2370}(41,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(-1\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{8}{13}\right)\)
\(\chi_{2370}(71,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(-1\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{7}{13}\right)\)
\(\chi_{2370}(191,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(-1\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{11}{13}\right)\)
\(\chi_{2370}(251,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(-1\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{23}{26}\right)\) \(e\left(\frac{4}{13}\right)\)
\(\chi_{2370}(491,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(-1\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{9}{13}\right)\)
\(\chi_{2370}(611,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(-1\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{17}{26}\right)\) \(e\left(\frac{12}{13}\right)\)
\(\chi_{2370}(701,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{1}{13}\right)\) \(-1\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{6}{13}\right)\)
\(\chi_{2370}(851,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{26}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{8}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(-1\) \(e\left(\frac{11}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{25}{26}\right)\) \(e\left(\frac{10}{13}\right)\)
\(\chi_{2370}(881,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(-1\) \(e\left(\frac{10}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{5}{26}\right)\) \(e\left(\frac{2}{13}\right)\)
\(\chi_{2370}(1121,\cdot)\) \(1\) \(1\) \(e\left(\frac{21}{26}\right)\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{6}{13}\right)\) \(e\left(\frac{11}{13}\right)\) \(-1\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(e\left(\frac{9}{26}\right)\) \(e\left(\frac{1}{13}\right)\)
\(\chi_{2370}(2081,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{3}{26}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{4}{13}\right)\) \(e\left(\frac{3}{13}\right)\) \(-1\) \(e\left(\frac{12}{13}\right)\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{19}{26}\right)\) \(e\left(\frac{5}{13}\right)\)
\(\chi_{2370}(2111,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{26}\right)\) \(e\left(\frac{7}{26}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{5}{13}\right)\) \(e\left(\frac{7}{13}\right)\) \(-1\) \(e\left(\frac{2}{13}\right)\) \(e\left(\frac{9}{13}\right)\) \(e\left(\frac{1}{26}\right)\) \(e\left(\frac{3}{13}\right)\)