↑

Properties

Label 3234.1973
Modulus $3234$
Conductor $1617$
Order $70$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

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([35,55,14]))
 
Copy content gp:[g,chi] = znchar(Mod(1973, 3234))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3234.1973");
 

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: \(1617\)
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 \(\chi_{1617}(356,\cdot)\)
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.cc

\(\chi_{3234}(125,\cdot)\) \(\chi_{3234}(251,\cdot)\) \(\chi_{3234}(335,\cdot)\) \(\chi_{3234}(377,\cdot)\) \(\chi_{3234}(713,\cdot)\) \(\chi_{3234}(797,\cdot)\) \(\chi_{3234}(839,\cdot)\) \(\chi_{3234}(1049,\cdot)\) \(\chi_{3234}(1259,\cdot)\) \(\chi_{3234}(1301,\cdot)\) \(\chi_{3234}(1511,\cdot)\) \(\chi_{3234}(1637,\cdot)\) \(\chi_{3234}(1721,\cdot)\) \(\chi_{3234}(1973,\cdot)\) \(\chi_{3234}(2099,\cdot)\) \(\chi_{3234}(2183,\cdot)\) \(\chi_{3234}(2225,\cdot)\) \(\chi_{3234}(2435,\cdot)\) \(\chi_{3234}(2561,\cdot)\) \(\chi_{3234}(2687,\cdot)\) \(\chi_{3234}(2897,\cdot)\) \(\chi_{3234}(3023,\cdot)\) \(\chi_{3234}(3107,\cdot)\) \(\chi_{3234}(3149,\cdot)\)

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

\((1079,199,2059)\) → \((-1,e\left(\frac{11}{14}\right),e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3234 }(1973, a) \) \(1\)\(1\)\(e\left(\frac{3}{35}\right)\)\(e\left(\frac{9}{70}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{6}{35}\right)\)\(e\left(\frac{3}{70}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{19}{35}\right)\)\(e\left(\frac{31}{35}\right)\)
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)
 
\( \chi_{ 3234 }(1973,a) \;\) at \(\;a = \) e.g. 2