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Properties

Label 2070.z
Modulus $2070$
Conductor $345$
Order $22$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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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(2070, base_ring=CyclotomicField(22)) M = H._module chi = DirichletCharacter(H, M([11,11,5])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(89, 2070)) 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("2070.89"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(2070\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(345\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(22\)
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 345.n
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: \(\Q(\zeta_{11})\)
Fixed field: Number field defined by a degree 22 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(7\) \(11\) \(13\) \(17\) \(19\) \(29\) \(31\) \(37\) \(41\) \(43\)
\(\chi_{2070}(89,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{7}{11}\right)\)
\(\chi_{2070}(359,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{3}{11}\right)\)
\(\chi_{2070}(539,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{2}{11}\right)\)
\(\chi_{2070}(1079,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{5}{11}\right)\)
\(\chi_{2070}(1169,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{10}{11}\right)\)
\(\chi_{2070}(1259,\cdot)\) \(1\) \(1\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{1}{11}\right)\)
\(\chi_{2070}(1349,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{4}{11}\right)\)
\(\chi_{2070}(1529,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{6}{11}\right)\)
\(\chi_{2070}(1709,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{9}{11}\right)\)
\(\chi_{2070}(1799,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{8}{11}\right)\)