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Properties

Label 1900.139
Modulus $1900$
Conductor $1900$
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(1900, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([45,27,70]))
 
Copy content gp:[g,chi] = znchar(Mod(139, 1900))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1900.139");
 

Basic properties

Modulus: \(1900\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1900\)
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 1900.ck

\(\chi_{1900}(119,\cdot)\) \(\chi_{1900}(139,\cdot)\) \(\chi_{1900}(339,\cdot)\) \(\chi_{1900}(359,\cdot)\) \(\chi_{1900}(479,\cdot)\) \(\chi_{1900}(519,\cdot)\) \(\chi_{1900}(579,\cdot)\) \(\chi_{1900}(719,\cdot)\) \(\chi_{1900}(739,\cdot)\) \(\chi_{1900}(859,\cdot)\) \(\chi_{1900}(879,\cdot)\) \(\chi_{1900}(959,\cdot)\) \(\chi_{1900}(1119,\cdot)\) \(\chi_{1900}(1239,\cdot)\) \(\chi_{1900}(1259,\cdot)\) \(\chi_{1900}(1279,\cdot)\) \(\chi_{1900}(1339,\cdot)\) \(\chi_{1900}(1479,\cdot)\) \(\chi_{1900}(1619,\cdot)\) \(\chi_{1900}(1639,\cdot)\) \(\chi_{1900}(1659,\cdot)\) \(\chi_{1900}(1719,\cdot)\) \(\chi_{1900}(1859,\cdot)\) \(\chi_{1900}(1879,\cdot)\)

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

\((951,77,401)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{7}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 1900 }(139, a) \) \(-1\)\(1\)\(e\left(\frac{32}{45}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{19}{45}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{53}{90}\right)\)\(e\left(\frac{61}{90}\right)\)\(e\left(\frac{17}{45}\right)\)\(e\left(\frac{16}{45}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{37}{45}\right)\)
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)
 
\( \chi_{ 1900 }(139,a) \;\) at \(\;a = \) e.g. 2