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Properties

Label 47096.9801
Modulus $47096$
Conductor $841$
Order $58$
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(47096, base_ring=CyclotomicField(58)) M = H._module chi = DirichletCharacter(H, M([0,0,0,49]))
 
Copy content gp:[g,chi] = znchar(Mod(9801, 47096))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("47096.9801");
 

Basic properties

Modulus: \(47096\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(841\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(58\)
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 \(\chi_{841}(550,\cdot)\)
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 47096.dv

\(\chi_{47096}(57,\cdot)\) \(\chi_{47096}(3305,\cdot)\) \(\chi_{47096}(4929,\cdot)\) \(\chi_{47096}(6553,\cdot)\) \(\chi_{47096}(8177,\cdot)\) \(\chi_{47096}(9801,\cdot)\) \(\chi_{47096}(11425,\cdot)\) \(\chi_{47096}(13049,\cdot)\) \(\chi_{47096}(14673,\cdot)\) \(\chi_{47096}(16297,\cdot)\) \(\chi_{47096}(17921,\cdot)\) \(\chi_{47096}(19545,\cdot)\) \(\chi_{47096}(21169,\cdot)\) \(\chi_{47096}(22793,\cdot)\) \(\chi_{47096}(24417,\cdot)\) \(\chi_{47096}(26041,\cdot)\) \(\chi_{47096}(27665,\cdot)\) \(\chi_{47096}(29289,\cdot)\) \(\chi_{47096}(30913,\cdot)\) \(\chi_{47096}(32537,\cdot)\) \(\chi_{47096}(34161,\cdot)\) \(\chi_{47096}(35785,\cdot)\) \(\chi_{47096}(37409,\cdot)\) \(\chi_{47096}(39033,\cdot)\) \(\chi_{47096}(40657,\cdot)\) \(\chi_{47096}(42281,\cdot)\) \(\chi_{47096}(43905,\cdot)\) \(\chi_{47096}(45529,\cdot)\)

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: $\Q(\zeta_{29})$
Fixed field: Number field defined by a degree 58 polynomial

Values on generators

\((11775,23549,13457,46257)\) → \((1,1,1,e\left(\frac{49}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 47096 }(9801, a) \) \(1\)\(1\)\(e\left(\frac{39}{58}\right)\)\(e\left(\frac{4}{29}\right)\)\(e\left(\frac{10}{29}\right)\)\(e\left(\frac{25}{58}\right)\)\(e\left(\frac{8}{29}\right)\)\(e\left(\frac{47}{58}\right)\)\(e\left(\frac{13}{58}\right)\)\(e\left(\frac{1}{58}\right)\)\(e\left(\frac{7}{29}\right)\)\(e\left(\frac{8}{29}\right)\)
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)
 
\( \chi_{ 47096 }(9801,a) \;\) at \(\;a = \) e.g. 2