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Properties

Label 4123.1077
Modulus $4123$
Conductor $4123$
Order $90$
Real no
Primitive yes
Minimal yes
Parity odd

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(4123, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([45,25,81]))
 
Copy content gp:[g,chi] = znchar(Mod(1077, 4123))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4123.1077");
 

Basic properties

Modulus: \(4123\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4123\)
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: 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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 4123.me

\(\chi_{4123}(356,\cdot)\) \(\chi_{4123}(573,\cdot)\) \(\chi_{4123}(678,\cdot)\) \(\chi_{4123}(895,\cdot)\) \(\chi_{4123}(1021,\cdot)\) \(\chi_{4123}(1077,\cdot)\) \(\chi_{4123}(1112,\cdot)\) \(\chi_{4123}(1238,\cdot)\) \(\chi_{4123}(1294,\cdot)\) \(\chi_{4123}(1511,\cdot)\) \(\chi_{4123}(1763,\cdot)\) \(\chi_{4123}(1875,\cdot)\) \(\chi_{4123}(2092,\cdot)\) \(\chi_{4123}(2162,\cdot)\) \(\chi_{4123}(2309,\cdot)\) \(\chi_{4123}(2540,\cdot)\) \(\chi_{4123}(2757,\cdot)\) \(\chi_{4123}(2960,\cdot)\) \(\chi_{4123}(2974,\cdot)\) \(\chi_{4123}(3282,\cdot)\) \(\chi_{4123}(3499,\cdot)\) \(\chi_{4123}(3625,\cdot)\) \(\chi_{4123}(3681,\cdot)\) \(\chi_{4123}(3898,\cdot)\)

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

\((1179,2605,1863)\) → \((-1,e\left(\frac{5}{18}\right),e\left(\frac{9}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 4123 }(1077, a) \) \(-1\)\(1\)\(e\left(\frac{79}{90}\right)\)\(e\left(\frac{1}{90}\right)\)\(e\left(\frac{34}{45}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{37}{45}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{23}{30}\right)\)
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)
 
\( \chi_{ 4123 }(1077,a) \;\) at \(\;a = \) e.g. 2