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Properties

Label 580.be
Modulus $580$
Conductor $580$
Order $28$
Real no
Primitive yes
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(580, base_ring=CyclotomicField(28)) M = H._module chi = DirichletCharacter(H, M([14,14,9])) chi.galois_orbit()
 
Copy content gp:[g,chi] = znchar(Mod(19, 580)) 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("580.19"); order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Basic properties

Modulus: \(580\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(580\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(28\)
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: yes
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: \(\Q(\zeta_{28})\)
Fixed field: 28.28.5002255640118107399365159746748272082794905600000000000000.1

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(7\) \(9\) \(11\) \(13\) \(17\) \(19\) \(21\) \(23\) \(27\)
\(\chi_{580}(19,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{2}{7}\right)\) \(i\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{23}{28}\right)\)
\(\chi_{580}(39,\cdot)\) \(1\) \(1\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{2}{7}\right)\) \(-i\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{9}{28}\right)\)
\(\chi_{580}(79,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{3}{7}\right)\) \(i\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{28}\right)\)
\(\chi_{580}(119,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{5}{7}\right)\) \(i\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{19}{28}\right)\)
\(\chi_{580}(159,\cdot)\) \(1\) \(1\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(i\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{27}{28}\right)\)
\(\chi_{580}(259,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{1}{7}\right)\) \(-i\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{28}\right)\)
\(\chi_{580}(279,\cdot)\) \(1\) \(1\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{4}{7}\right)\) \(-i\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{25}{28}\right)\)
\(\chi_{580}(359,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{4}{7}\right)\) \(i\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{11}{28}\right)\)
\(\chi_{580}(379,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{1}{7}\right)\) \(i\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{15}{28}\right)\)
\(\chi_{580}(479,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{6}{7}\right)\) \(-i\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{13}{28}\right)\)
\(\chi_{580}(519,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{5}{7}\right)\) \(-i\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{28}\right)\)
\(\chi_{580}(559,\cdot)\) \(1\) \(1\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{3}{7}\right)\) \(-i\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{17}{28}\right)\)