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Properties

Label 97682.16811
Modulus $97682$
Conductor $3757$
Order $408$
Real no
Primitive no
Minimal no
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(97682, base_ring=CyclotomicField(408)) M = H._module chi = DirichletCharacter(H, M([34,57]))
 
Copy content gp:[g,chi] = znchar(Mod(16811, 97682))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("97682.16811");
 

Basic properties

Modulus: \(97682\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3757\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(408\)
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 \(\chi_{3757}(1783,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 97682.dw

\(\chi_{97682}(19,\cdot)\) \(\chi_{97682}(695,\cdot)\) \(\chi_{97682}(995,\cdot)\) \(\chi_{97682}(2117,\cdot)\) \(\chi_{97682}(4037,\cdot)\) \(\chi_{97682}(4643,\cdot)\) \(\chi_{97682}(5159,\cdot)\) \(\chi_{97682}(5319,\cdot)\) \(\chi_{97682}(5765,\cdot)\) \(\chi_{97682}(6441,\cdot)\) \(\chi_{97682}(6741,\cdot)\) \(\chi_{97682}(7863,\cdot)\) \(\chi_{97682}(9783,\cdot)\) \(\chi_{97682}(10389,\cdot)\) \(\chi_{97682}(10905,\cdot)\) \(\chi_{97682}(11065,\cdot)\) \(\chi_{97682}(11511,\cdot)\) \(\chi_{97682}(12187,\cdot)\) \(\chi_{97682}(12487,\cdot)\) \(\chi_{97682}(13609,\cdot)\) \(\chi_{97682}(15529,\cdot)\) \(\chi_{97682}(16135,\cdot)\) \(\chi_{97682}(16651,\cdot)\) \(\chi_{97682}(16811,\cdot)\) \(\chi_{97682}(17257,\cdot)\) \(\chi_{97682}(17933,\cdot)\) \(\chi_{97682}(18233,\cdot)\) \(\chi_{97682}(19355,\cdot)\) \(\chi_{97682}(21275,\cdot)\) \(\chi_{97682}(21881,\cdot)\) ...

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: $\Q(\zeta_{408})$
Fixed field: Number field defined by a degree 408 polynomial (not computed)

Values on generators

\((28901,39885)\) → \((e\left(\frac{1}{12}\right),e\left(\frac{19}{136}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(19\)\(21\)\(23\)\(25\)
\( \chi_{ 97682 }(16811, a) \) \(-1\)\(1\)\(e\left(\frac{193}{408}\right)\)\(e\left(\frac{101}{136}\right)\)\(e\left(\frac{233}{408}\right)\)\(e\left(\frac{193}{204}\right)\)\(e\left(\frac{325}{408}\right)\)\(e\left(\frac{11}{51}\right)\)\(e\left(\frac{19}{51}\right)\)\(e\left(\frac{3}{68}\right)\)\(e\left(\frac{403}{408}\right)\)\(e\left(\frac{33}{68}\right)\)
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)
 
\( \chi_{ 97682 }(16811,a) \;\) at \(\;a = \) e.g. 2