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Properties

Label 305045.20257
Modulus $305045$
Conductor $23465$
Order $684$
Real no
Primitive no
Minimal no
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(305045, base_ring=CyclotomicField(684)) M = H._module chi = DirichletCharacter(H, M([171,228,134]))
 
Copy content gp:[g,chi] = znchar(Mod(20257, 305045))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("305045.20257");
 

Basic properties

Modulus: \(305045\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(23465\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(684\)
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 \(\chi_{23465}(20257,\cdot)\)
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: no
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 305045.rq

\(\chi_{305045}(1667,\cdot)\) \(\chi_{305045}(2388,\cdot)\) \(\chi_{305045}(4202,\cdot)\) \(\chi_{305045}(4247,\cdot)\) \(\chi_{305045}(4878,\cdot)\) \(\chi_{305045}(7413,\cdot)\) \(\chi_{305045}(7458,\cdot)\) \(\chi_{305045}(9103,\cdot)\) \(\chi_{305045}(11852,\cdot)\) \(\chi_{305045}(15232,\cdot)\) \(\chi_{305045}(17722,\cdot)\) \(\chi_{305045}(18443,\cdot)\) \(\chi_{305045}(20257,\cdot)\) \(\chi_{305045}(20302,\cdot)\) \(\chi_{305045}(20933,\cdot)\) \(\chi_{305045}(21947,\cdot)\) \(\chi_{305045}(23468,\cdot)\) \(\chi_{305045}(23513,\cdot)\) \(\chi_{305045}(25158,\cdot)\) \(\chi_{305045}(27907,\cdot)\) \(\chi_{305045}(31118,\cdot)\) \(\chi_{305045}(31287,\cdot)\) \(\chi_{305045}(33777,\cdot)\) \(\chi_{305045}(34498,\cdot)\) \(\chi_{305045}(36312,\cdot)\) \(\chi_{305045}(36357,\cdot)\) \(\chi_{305045}(36988,\cdot)\) \(\chi_{305045}(38002,\cdot)\) \(\chi_{305045}(39523,\cdot)\) \(\chi_{305045}(39568,\cdot)\) ...

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: $\Q(\zeta_{684})$
Fixed field: Number field defined by a degree 684 polynomial (not computed)

Values on generators

\((244037,175086,190971)\) → \((i,e\left(\frac{1}{3}\right),e\left(\frac{67}{342}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\( \chi_{ 305045 }(20257, a) \) \(1\)\(1\)\(e\left(\frac{533}{684}\right)\)\(e\left(\frac{215}{684}\right)\)\(e\left(\frac{191}{342}\right)\)\(e\left(\frac{16}{171}\right)\)\(e\left(\frac{23}{76}\right)\)\(e\left(\frac{77}{228}\right)\)\(e\left(\frac{215}{342}\right)\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{199}{228}\right)\)\(e\left(\frac{14}{171}\right)\)
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)
 
\( \chi_{ 305045 }(20257,a) \;\) at \(\;a = \) e.g. 2