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Properties

Label 3234.1399
Modulus $3234$
Conductor $539$
Order $70$
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(3234, base_ring=CyclotomicField(70)) M = H._module chi = DirichletCharacter(H, M([0,5,7]))
 
Copy content gp:[g,chi] = znchar(Mod(1399, 3234))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3234.1399");
 

Basic properties

Modulus: \(3234\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(539\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(70\)
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 \(\chi_{539}(321,\cdot)\)
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 3234.ca

\(\chi_{3234}(13,\cdot)\) \(\chi_{3234}(139,\cdot)\) \(\chi_{3234}(349,\cdot)\) \(\chi_{3234}(475,\cdot)\) \(\chi_{3234}(601,\cdot)\) \(\chi_{3234}(811,\cdot)\) \(\chi_{3234}(853,\cdot)\) \(\chi_{3234}(937,\cdot)\) \(\chi_{3234}(1063,\cdot)\) \(\chi_{3234}(1315,\cdot)\) \(\chi_{3234}(1399,\cdot)\) \(\chi_{3234}(1525,\cdot)\) \(\chi_{3234}(1735,\cdot)\) \(\chi_{3234}(1777,\cdot)\) \(\chi_{3234}(1987,\cdot)\) \(\chi_{3234}(2197,\cdot)\) \(\chi_{3234}(2239,\cdot)\) \(\chi_{3234}(2323,\cdot)\) \(\chi_{3234}(2659,\cdot)\) \(\chi_{3234}(2701,\cdot)\) \(\chi_{3234}(2785,\cdot)\) \(\chi_{3234}(2911,\cdot)\) \(\chi_{3234}(3121,\cdot)\) \(\chi_{3234}(3163,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((1079,199,2059)\) → \((1,e\left(\frac{1}{14}\right),e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3234 }(1399, a) \) \(1\)\(1\)\(e\left(\frac{33}{70}\right)\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{69}{70}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{13}{35}\right)\)
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)
 
\( \chi_{ 3234 }(1399,a) \;\) at \(\;a = \) e.g. 2