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Properties

Label 142175.16012
Modulus $142175$
Conductor $12925$
Order $460$
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(142175, base_ring=CyclotomicField(460)) M = H._module chi = DirichletCharacter(H, M([207,322,440]))
 
Copy content gp:[g,chi] = znchar(Mod(16012, 142175))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("142175.16012");
 

Basic properties

Modulus: \(142175\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(12925\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(460\)
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 \(\chi_{12925}(3087,\cdot)\)
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 142175.je

\(\chi_{142175}(723,\cdot)\) \(\chi_{142175}(1183,\cdot)\) \(\chi_{142175}(1322,\cdot)\) \(\chi_{142175}(2653,\cdot)\) \(\chi_{142175}(3627,\cdot)\) \(\chi_{142175}(4208,\cdot)\) \(\chi_{142175}(4638,\cdot)\) \(\chi_{142175}(6652,\cdot)\) \(\chi_{142175}(6937,\cdot)\) \(\chi_{142175}(7233,\cdot)\) \(\chi_{142175}(7663,\cdot)\) \(\chi_{142175}(7717,\cdot)\) \(\chi_{142175}(8703,\cdot)\) \(\chi_{142175}(9677,\cdot)\) \(\chi_{142175}(10258,\cdot)\) \(\chi_{142175}(11728,\cdot)\) \(\chi_{142175}(12702,\cdot)\) \(\chi_{142175}(13422,\cdot)\) \(\chi_{142175}(13713,\cdot)\) \(\chi_{142175}(14753,\cdot)\) \(\chi_{142175}(15848,\cdot)\) \(\chi_{142175}(16012,\cdot)\) \(\chi_{142175}(16738,\cdot)\) \(\chi_{142175}(17778,\cdot)\) \(\chi_{142175}(19037,\cdot)\) \(\chi_{142175}(19333,\cdot)\) \(\chi_{142175}(19472,\cdot)\) \(\chi_{142175}(21777,\cdot)\) \(\chi_{142175}(25087,\cdot)\) \(\chi_{142175}(25383,\cdot)\) ...

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: $\Q(\zeta_{460})$
Fixed field: Number field defined by a degree 460 polynomial (not computed)

Values on generators

\((130802,44651,9076)\) → \((e\left(\frac{9}{20}\right),e\left(\frac{7}{10}\right),e\left(\frac{22}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 142175 }(16012, a) \) \(1\)\(1\)\(e\left(\frac{169}{460}\right)\)\(e\left(\frac{81}{92}\right)\)\(e\left(\frac{169}{230}\right)\)\(e\left(\frac{57}{230}\right)\)\(e\left(\frac{349}{460}\right)\)\(e\left(\frac{47}{460}\right)\)\(e\left(\frac{35}{46}\right)\)\(e\left(\frac{283}{460}\right)\)\(e\left(\frac{71}{92}\right)\)\(e\left(\frac{29}{230}\right)\)
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)
 
\( \chi_{ 142175 }(16012,a) \;\) at \(\;a = \) e.g. 2