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Properties

Label 238521.1297
Modulus $238521$
Conductor $79507$
Order $11094$
Real no
Primitive no
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(238521, base_ring=CyclotomicField(11094)) M = H._module chi = DirichletCharacter(H, M([0,1559]))
 
Copy content gp:[g,chi] = znchar(Mod(1297, 238521))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("238521.1297");
 

Basic properties

Modulus: \(238521\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(79507\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(11094\)
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 \(\chi_{79507}(1297,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 238521.bm

\(\chi_{238521}(7,\cdot)\) \(\chi_{238521}(37,\cdot)\) \(\chi_{238521}(136,\cdot)\) \(\chi_{238521}(166,\cdot)\) \(\chi_{238521}(265,\cdot)\) \(\chi_{238521}(295,\cdot)\) \(\chi_{238521}(394,\cdot)\) \(\chi_{238521}(523,\cdot)\) \(\chi_{238521}(553,\cdot)\) \(\chi_{238521}(652,\cdot)\) \(\chi_{238521}(682,\cdot)\) \(\chi_{238521}(781,\cdot)\) \(\chi_{238521}(811,\cdot)\) \(\chi_{238521}(910,\cdot)\) \(\chi_{238521}(940,\cdot)\) \(\chi_{238521}(1039,\cdot)\) \(\chi_{238521}(1069,\cdot)\) \(\chi_{238521}(1168,\cdot)\) \(\chi_{238521}(1198,\cdot)\) \(\chi_{238521}(1297,\cdot)\) \(\chi_{238521}(1327,\cdot)\) \(\chi_{238521}(1456,\cdot)\)

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: $\Q(\zeta_{5547})$
Fixed field: Number field defined by a degree 11094 polynomial (not computed)

Values on generators

\((79508,79510)\) → \((1,e\left(\frac{1559}{11094}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 238521 }(1297, a) \) \(-1\)\(1\)\(e\left(\frac{3321}{3698}\right)\)\(e\left(\frac{1472}{1849}\right)\)\(e\left(\frac{1091}{11094}\right)\)\(e\left(\frac{5017}{11094}\right)\)\(e\left(\frac{2567}{3698}\right)\)\(e\left(\frac{5527}{5547}\right)\)\(e\left(\frac{1512}{1849}\right)\)\(e\left(\frac{3656}{5547}\right)\)\(e\left(\frac{1943}{5547}\right)\)\(e\left(\frac{1095}{1849}\right)\)
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)
 
\( \chi_{ 238521 }(1297,a) \;\) at \(\;a = \) e.g. 2