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Properties

Label 3708.1903
Modulus $3708$
Conductor $3708$
Order $102$
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(3708, base_ring=CyclotomicField(102)) M = H._module chi = DirichletCharacter(H, M([51,34,8]))
 
Copy content gp:[g,chi] = znchar(Mod(1903, 3708))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3708.1903");
 

Basic properties

Modulus: \(3708\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3708\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(102\)
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 3708.cx

\(\chi_{3708}(7,\cdot)\) \(\chi_{3708}(223,\cdot)\) \(\chi_{3708}(247,\cdot)\) \(\chi_{3708}(367,\cdot)\) \(\chi_{3708}(475,\cdot)\) \(\chi_{3708}(547,\cdot)\) \(\chi_{3708}(643,\cdot)\) \(\chi_{3708}(715,\cdot)\) \(\chi_{3708}(979,\cdot)\) \(\chi_{3708}(1231,\cdot)\) \(\chi_{3708}(1255,\cdot)\) \(\chi_{3708}(1291,\cdot)\) \(\chi_{3708}(1375,\cdot)\) \(\chi_{3708}(1399,\cdot)\) \(\chi_{3708}(1471,\cdot)\) \(\chi_{3708}(1627,\cdot)\) \(\chi_{3708}(1663,\cdot)\) \(\chi_{3708}(1843,\cdot)\) \(\chi_{3708}(1903,\cdot)\) \(\chi_{3708}(1975,\cdot)\) \(\chi_{3708}(2119,\cdot)\) \(\chi_{3708}(2167,\cdot)\) \(\chi_{3708}(2191,\cdot)\) \(\chi_{3708}(2371,\cdot)\) \(\chi_{3708}(2419,\cdot)\) \(\chi_{3708}(2563,\cdot)\) \(\chi_{3708}(3055,\cdot)\) \(\chi_{3708}(3379,\cdot)\) \(\chi_{3708}(3415,\cdot)\) \(\chi_{3708}(3535,\cdot)\) ...

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: $\Q(\zeta_{51})$
Fixed field: Number field defined by a degree 102 polynomial (not computed)

Values on generators

\((1855,1649,829)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{4}{51}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 3708 }(1903, a) \) \(-1\)\(1\)\(e\left(\frac{38}{51}\right)\)\(e\left(\frac{5}{34}\right)\)\(e\left(\frac{21}{34}\right)\)\(e\left(\frac{16}{51}\right)\)\(e\left(\frac{25}{51}\right)\)\(e\left(\frac{79}{102}\right)\)\(e\left(\frac{5}{102}\right)\)\(e\left(\frac{25}{51}\right)\)\(e\left(\frac{4}{51}\right)\)\(e\left(\frac{65}{102}\right)\)
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)
 
\( \chi_{ 3708 }(1903,a) \;\) at \(\;a = \) e.g. 2