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Properties

Label 6650.519
Modulus $6650$
Conductor $475$
Order $90$
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
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6650, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([81,0,70]))
 
Copy content gp:[g,chi] = znchar(Mod(519, 6650))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6650.519");
 

Basic properties

Modulus: \(6650\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(475\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(90\)
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 \(\chi_{475}(44,\cdot)\)
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);
 

Galois orbit 6650.gp

\(\chi_{6650}(169,\cdot)\) \(\chi_{6650}(309,\cdot)\) \(\chi_{6650}(519,\cdot)\) \(\chi_{6650}(1289,\cdot)\) \(\chi_{6650}(1429,\cdot)\) \(\chi_{6650}(1639,\cdot)\) \(\chi_{6650}(2479,\cdot)\) \(\chi_{6650}(2619,\cdot)\) \(\chi_{6650}(2759,\cdot)\) \(\chi_{6650}(2829,\cdot)\) \(\chi_{6650}(2969,\cdot)\) \(\chi_{6650}(3179,\cdot)\) \(\chi_{6650}(3809,\cdot)\) \(\chi_{6650}(4089,\cdot)\) \(\chi_{6650}(4159,\cdot)\) \(\chi_{6650}(4509,\cdot)\) \(\chi_{6650}(5139,\cdot)\) \(\chi_{6650}(5279,\cdot)\) \(\chi_{6650}(5419,\cdot)\) \(\chi_{6650}(5489,\cdot)\) \(\chi_{6650}(5629,\cdot)\) \(\chi_{6650}(5839,\cdot)\) \(\chi_{6650}(6469,\cdot)\) \(\chi_{6650}(6609,\cdot)\)

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((2927,5701,4201)\) → \((e\left(\frac{9}{10}\right),1,e\left(\frac{7}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(23\)\(27\)\(29\)\(31\)\(33\)
\( \chi_{ 6650 }(519, a) \) \(1\)\(1\)\(e\left(\frac{37}{90}\right)\)\(e\left(\frac{37}{45}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{89}{90}\right)\)\(e\left(\frac{43}{90}\right)\)\(e\left(\frac{41}{90}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{13}{90}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 6650 }(519,a) \;\) at \(\;a = \) e.g. 2