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Properties

Label 17168.1513
Modulus $17168$
Conductor $8584$
Order $126$
Real no
Primitive no
Minimal no
Parity even

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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(17168, base_ring=CyclotomicField(126)) M = H._module chi = DirichletCharacter(H, M([0,63,99,70]))
 
Copy content gp:[g,chi] = znchar(Mod(1513, 17168))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("17168.1513");
 

Basic properties

Modulus: \(17168\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(8584\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(126\)
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 \(\chi_{8584}(5805,\cdot)\)
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 17168.ok

\(\chi_{17168}(9,\cdot)\) \(\chi_{17168}(441,\cdot)\) \(\chi_{17168}(921,\cdot)\) \(\chi_{17168}(937,\cdot)\) \(\chi_{17168}(1193,\cdot)\) \(\chi_{17168}(1385,\cdot)\) \(\chi_{17168}(1513,\cdot)\) \(\chi_{17168}(1977,\cdot)\) \(\chi_{17168}(2121,\cdot)\) \(\chi_{17168}(2217,\cdot)\) \(\chi_{17168}(4153,\cdot)\) \(\chi_{17168}(5081,\cdot)\) \(\chi_{17168}(5113,\cdot)\) \(\chi_{17168}(5929,\cdot)\) \(\chi_{17168}(6297,\cdot)\) \(\chi_{17168}(6857,\cdot)\) \(\chi_{17168}(6953,\cdot)\) \(\chi_{17168}(7433,\cdot)\) \(\chi_{17168}(7545,\cdot)\) \(\chi_{17168}(7897,\cdot)\) \(\chi_{17168}(8617,\cdot)\) \(\chi_{17168}(9081,\cdot)\) \(\chi_{17168}(9257,\cdot)\) \(\chi_{17168}(10665,\cdot)\) \(\chi_{17168}(11033,\cdot)\) \(\chi_{17168}(11257,\cdot)\) \(\chi_{17168}(11577,\cdot)\) \(\chi_{17168}(11593,\cdot)\) \(\chi_{17168}(12041,\cdot)\) \(\chi_{17168}(12185,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

\((15023,4293,10065,6033)\) → \((1,-1,e\left(\frac{11}{14}\right),e\left(\frac{5}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 17168 }(1513, a) \) \(1\)\(1\)\(e\left(\frac{55}{63}\right)\)\(e\left(\frac{71}{126}\right)\)\(e\left(\frac{13}{63}\right)\)\(e\left(\frac{47}{63}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{95}{126}\right)\)\(e\left(\frac{55}{126}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{1}{63}\right)\)\(e\left(\frac{5}{63}\right)\)
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)
 
\( \chi_{ 17168 }(1513,a) \;\) at \(\;a = \) e.g. 2