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Properties

Label 16362.5591
Modulus $16362$
Conductor $8181$
Order $270$
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(16362, base_ring=CyclotomicField(270)) M = H._module chi = DirichletCharacter(H, M([5,108]))
 
Copy content gp:[g,chi] = znchar(Mod(5591, 16362))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("16362.5591");
 

Basic properties

Modulus: \(16362\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(8181\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(270\)
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 \(\chi_{8181}(5591,\cdot)\)
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 16362.cd

\(\chi_{16362}(95,\cdot)\) \(\chi_{16362}(137,\cdot)\) \(\chi_{16362}(185,\cdot)\) \(\chi_{16362}(491,\cdot)\) \(\chi_{16362}(743,\cdot)\) \(\chi_{16362}(1307,\cdot)\) \(\chi_{16362}(1397,\cdot)\) \(\chi_{16362}(1703,\cdot)\) \(\chi_{16362}(1913,\cdot)\) \(\chi_{16362}(1955,\cdot)\) \(\chi_{16362}(2003,\cdot)\) \(\chi_{16362}(2309,\cdot)\) \(\chi_{16362}(2561,\cdot)\) \(\chi_{16362}(3125,\cdot)\) \(\chi_{16362}(3215,\cdot)\) \(\chi_{16362}(3521,\cdot)\) \(\chi_{16362}(3731,\cdot)\) \(\chi_{16362}(3773,\cdot)\) \(\chi_{16362}(3821,\cdot)\) \(\chi_{16362}(4127,\cdot)\) \(\chi_{16362}(4379,\cdot)\) \(\chi_{16362}(4943,\cdot)\) \(\chi_{16362}(5033,\cdot)\) \(\chi_{16362}(5339,\cdot)\) \(\chi_{16362}(5549,\cdot)\) \(\chi_{16362}(5591,\cdot)\) \(\chi_{16362}(5639,\cdot)\) \(\chi_{16362}(5945,\cdot)\) \(\chi_{16362}(6197,\cdot)\) \(\chi_{16362}(6761,\cdot)\) ...

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: $\Q(\zeta_{135})$
Fixed field: Number field defined by a degree 270 polynomial (not computed)

Values on generators

\((15959,8587)\) → \((e\left(\frac{1}{54}\right),e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 16362 }(5591, a) \) \(-1\)\(1\)\(e\left(\frac{7}{270}\right)\)\(e\left(\frac{121}{135}\right)\)\(e\left(\frac{119}{270}\right)\)\(e\left(\frac{74}{135}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{13}{45}\right)\)\(e\left(\frac{163}{270}\right)\)\(e\left(\frac{7}{135}\right)\)\(e\left(\frac{23}{270}\right)\)\(e\left(\frac{131}{135}\right)\)
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)
 
\( \chi_{ 16362 }(5591,a) \;\) at \(\;a = \) e.g. 2