↑

Properties

Label 47275.33253
Modulus $47275$
Conductor $47275$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

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(47275, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([21,58,3]))
 
Copy content gp:[g,chi] = znchar(Mod(33253, 47275))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("47275.33253");
 

Basic properties

Modulus: \(47275\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(47275\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(60\)
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 47275.cqm

\(\chi_{47275}(2163,\cdot)\) \(\chi_{47275}(3148,\cdot)\) \(\chi_{47275}(6738,\cdot)\) \(\chi_{47275}(7328,\cdot)\) \(\chi_{47275}(10037,\cdot)\) \(\chi_{47275}(12298,\cdot)\) \(\chi_{47275}(14612,\cdot)\) \(\chi_{47275}(21083,\cdot)\) \(\chi_{47275}(21252,\cdot)\) \(\chi_{47275}(24697,\cdot)\) \(\chi_{47275}(27717,\cdot)\) \(\chi_{47275}(30402,\cdot)\) \(\chi_{47275}(33253,\cdot)\) \(\chi_{47275}(39383,\cdot)\) \(\chi_{47275}(46017,\cdot)\) \(\chi_{47275}(46047,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((20802,32026,34101)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{29}{30}\right),e\left(\frac{1}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 47275 }(33253, a) \) \(-1\)\(1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{17}{60}\right)\)
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)
 
\( \chi_{ 47275 }(33253,a) \;\) at \(\;a = \) e.g. 2