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Properties

Label 125341.1340
Modulus $125341$
Conductor $125341$
Order $3600$
Real no
Primitive yes
Minimal yes
Parity odd

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(125341, base_ring=CyclotomicField(3600)) M = H._module chi = DirichletCharacter(H, M([2025,3350,252]))
 
Copy content gp:[g,chi] = znchar(Mod(1340, 125341))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("125341.1340");
 

Basic properties

Modulus: \(125341\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(125341\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(3600\)
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 125341.bnu

\(\chi_{125341}(99,\cdot)\) \(\chi_{125341}(112,\cdot)\) \(\chi_{125341}(160,\cdot)\) \(\chi_{125341}(351,\cdot)\) \(\chi_{125341}(354,\cdot)\) \(\chi_{125341}(452,\cdot)\) \(\chi_{125341}(498,\cdot)\) \(\chi_{125341}(823,\cdot)\) \(\chi_{125341}(843,\cdot)\) \(\chi_{125341}(856,\cdot)\) \(\chi_{125341}(938,\cdot)\) \(\chi_{125341}(1108,\cdot)\) \(\chi_{125341}(1153,\cdot)\) \(\chi_{125341}(1286,\cdot)\) \(\chi_{125341}(1340,\cdot)\)

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: $\Q(\zeta_{3600})$
Fixed field: Number field defined by a degree 3600 polynomial (not computed)

Values on generators

\((22120,46360,8688)\) → \((e\left(\frac{9}{16}\right),e\left(\frac{67}{72}\right),e\left(\frac{7}{100}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 125341 }(1340, a) \) \(-1\)\(1\)\(e\left(\frac{701}{1800}\right)\)\(e\left(\frac{1171}{1200}\right)\)\(e\left(\frac{701}{900}\right)\)\(e\left(\frac{1523}{3600}\right)\)\(e\left(\frac{263}{720}\right)\)\(e\left(\frac{631}{1200}\right)\)\(e\left(\frac{101}{600}\right)\)\(e\left(\frac{571}{600}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{101}{3600}\right)\)
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)
 
\( \chi_{ 125341 }(1340,a) \;\) at \(\;a = \) e.g. 2