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Properties

Label 3234.1459
Modulus $3234$
Conductor $539$
Order $210$
Real no
Primitive no
Minimal yes
Parity even

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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(210)) M = H._module chi = DirichletCharacter(H, M([0,95,147]))
 
Copy content gp:[g,chi] = znchar(Mod(1459, 3234))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3234.1459");
 

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: \(210\)
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 \(\chi_{539}(381,\cdot)\)
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.ck

\(\chi_{3234}(61,\cdot)\) \(\chi_{3234}(73,\cdot)\) \(\chi_{3234}(145,\cdot)\) \(\chi_{3234}(271,\cdot)\) \(\chi_{3234}(283,\cdot)\) \(\chi_{3234}(409,\cdot)\) \(\chi_{3234}(481,\cdot)\) \(\chi_{3234}(523,\cdot)\) \(\chi_{3234}(535,\cdot)\) \(\chi_{3234}(733,\cdot)\) \(\chi_{3234}(745,\cdot)\) \(\chi_{3234}(787,\cdot)\) \(\chi_{3234}(871,\cdot)\) \(\chi_{3234}(943,\cdot)\) \(\chi_{3234}(985,\cdot)\) \(\chi_{3234}(997,\cdot)\) \(\chi_{3234}(1069,\cdot)\) \(\chi_{3234}(1249,\cdot)\) \(\chi_{3234}(1333,\cdot)\) \(\chi_{3234}(1405,\cdot)\) \(\chi_{3234}(1447,\cdot)\) \(\chi_{3234}(1459,\cdot)\) \(\chi_{3234}(1531,\cdot)\) \(\chi_{3234}(1657,\cdot)\) \(\chi_{3234}(1669,\cdot)\) \(\chi_{3234}(1711,\cdot)\) \(\chi_{3234}(1867,\cdot)\) \(\chi_{3234}(1909,\cdot)\) \(\chi_{3234}(1921,\cdot)\) \(\chi_{3234}(1993,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial (not computed)

Values on generators

\((1079,199,2059)\) → \((1,e\left(\frac{19}{42}\right),e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3234 }(1459, a) \) \(1\)\(1\)\(e\left(\frac{193}{210}\right)\)\(e\left(\frac{22}{35}\right)\)\(e\left(\frac{64}{105}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{88}{105}\right)\)\(e\left(\frac{3}{70}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{92}{105}\right)\)\(e\left(\frac{31}{35}\right)\)
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)
 
\( \chi_{ 3234 }(1459,a) \;\) at \(\;a = \) e.g. 2