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Properties

Label 23698.217
Modulus $23698$
Conductor $11849$
Order $680$
Real no
Primitive no
Minimal yes
Parity odd

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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(23698, base_ring=CyclotomicField(680)) M = H._module chi = DirichletCharacter(H, M([410,459]))
 
Copy content gp:[g,chi] = znchar(Mod(217, 23698))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("23698.217");
 

Basic properties

Modulus: \(23698\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11849\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(680\)
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 \(\chi_{11849}(217,\cdot)\)
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 23698.dx

\(\chi_{23698}(13,\cdot)\) \(\chi_{23698}(217,\cdot)\) \(\chi_{23698}(293,\cdot)\) \(\chi_{23698}(395,\cdot)\) \(\chi_{23698}(421,\cdot)\) \(\chi_{23698}(429,\cdot)\) \(\chi_{23698}(727,\cdot)\) \(\chi_{23698}(803,\cdot)\) \(\chi_{23698}(837,\cdot)\) \(\chi_{23698}(931,\cdot)\) \(\chi_{23698}(1135,\cdot)\) \(\chi_{23698}(1211,\cdot)\) \(\chi_{23698}(1237,\cdot)\) \(\chi_{23698}(1245,\cdot)\) \(\chi_{23698}(1305,\cdot)\) \(\chi_{23698}(1347,\cdot)\) \(\chi_{23698}(1611,\cdot)\) \(\chi_{23698}(1687,\cdot)\) \(\chi_{23698}(1789,\cdot)\) \(\chi_{23698}(1815,\cdot)\) \(\chi_{23698}(1823,\cdot)\) \(\chi_{23698}(2121,\cdot)\) \(\chi_{23698}(2197,\cdot)\) \(\chi_{23698}(2231,\cdot)\) \(\chi_{23698}(2325,\cdot)\) \(\chi_{23698}(2529,\cdot)\) \(\chi_{23698}(2605,\cdot)\) \(\chi_{23698}(2631,\cdot)\) \(\chi_{23698}(2699,\cdot)\) \(\chi_{23698}(2741,\cdot)\) ...

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: $\Q(\zeta_{680})$
Fixed field: Number field defined by a degree 680 polynomial (not computed)

Values on generators

\((11563,18497)\) → \((e\left(\frac{41}{68}\right),e\left(\frac{27}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(19\)\(21\)\(23\)
\( \chi_{ 23698 }(217, a) \) \(-1\)\(1\)\(e\left(\frac{99}{136}\right)\)\(e\left(\frac{157}{170}\right)\)\(e\left(\frac{531}{680}\right)\)\(e\left(\frac{31}{68}\right)\)\(e\left(\frac{607}{680}\right)\)\(e\left(\frac{69}{680}\right)\)\(e\left(\frac{443}{680}\right)\)\(e\left(\frac{351}{680}\right)\)\(e\left(\frac{173}{340}\right)\)\(e\left(\frac{257}{340}\right)\)
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)
 
\( \chi_{ 23698 }(217,a) \;\) at \(\;a = \) e.g. 2