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Properties

Label 47430.761
Modulus $47430$
Conductor $4743$
Order $60$
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(47430, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([50,0,15,14]))
 
Copy content gp:[g,chi] = znchar(Mod(761, 47430))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("47430.761");
 

Basic properties

Modulus: \(47430\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4743\)
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: no, induced from \(\chi_{4743}(761,\cdot)\)
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 47430.ty

\(\chi_{47430}(761,\cdot)\) \(\chi_{47430}(2801,\cdot)\) \(\chi_{47430}(4271,\cdot)\) \(\chi_{47430}(4331,\cdot)\) \(\chi_{47430}(7391,\cdot)\) \(\chi_{47430}(11921,\cdot)\) \(\chi_{47430}(13001,\cdot)\) \(\chi_{47430}(13961,\cdot)\) \(\chi_{47430}(15491,\cdot)\) \(\chi_{47430}(16061,\cdot)\) \(\chi_{47430}(18551,\cdot)\) \(\chi_{47430}(24161,\cdot)\) \(\chi_{47430}(25751,\cdot)\) \(\chi_{47430}(27221,\cdot)\) \(\chi_{47430}(36911,\cdot)\) \(\chi_{47430}(40541,\cdot)\)

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

\((10541,9487,2791,4591)\) → \((e\left(\frac{5}{6}\right),1,i,e\left(\frac{7}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(19\)\(23\)\(29\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 47430 }(761, a) \) \(1\)\(1\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{7}{30}\right)\)
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)
 
\( \chi_{ 47430 }(761,a) \;\) at \(\;a = \) e.g. 2