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Properties

Label 32718.17363
Modulus $32718$
Conductor $16359$
Order $180$
Real no
Primitive no
Minimal yes
Parity even

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(32718, base_ring=CyclotomicField(180)) M = H._module chi = DirichletCharacter(H, M([90,30,40,153]))
 
Copy content gp:[g,chi] = znchar(Mod(17363, 32718))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("32718.17363");
 

Basic properties

Modulus: \(32718\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(16359\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(180\)
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 \(\chi_{16359}(1004,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 32718.ss

\(\chi_{32718}(689,\cdot)\) \(\chi_{32718}(815,\cdot)\) \(\chi_{32718}(2411,\cdot)\) \(\chi_{32718}(3083,\cdot)\) \(\chi_{32718}(4679,\cdot)\) \(\chi_{32718}(4805,\cdot)\) \(\chi_{32718}(5801,\cdot)\) \(\chi_{32718}(6275,\cdot)\) \(\chi_{32718}(6401,\cdot)\) \(\chi_{32718}(6599,\cdot)\) \(\chi_{32718}(7073,\cdot)\) \(\chi_{32718}(7997,\cdot)\) \(\chi_{32718}(8195,\cdot)\) \(\chi_{32718}(8795,\cdot)\) \(\chi_{32718}(9791,\cdot)\) \(\chi_{32718}(10967,\cdot)\) \(\chi_{32718}(11765,\cdot)\) \(\chi_{32718}(12185,\cdot)\) \(\chi_{32718}(13361,\cdot)\) \(\chi_{32718}(13781,\cdot)\) \(\chi_{32718}(14411,\cdot)\) \(\chi_{32718}(14957,\cdot)\) \(\chi_{32718}(15209,\cdot)\) \(\chi_{32718}(15377,\cdot)\) \(\chi_{32718}(16175,\cdot)\) \(\chi_{32718}(16805,\cdot)\) \(\chi_{32718}(17351,\cdot)\) \(\chi_{32718}(17363,\cdot)\) \(\chi_{32718}(18161,\cdot)\) \(\chi_{32718}(18401,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((10907,28045,13777,19153)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{2}{9}\right),e\left(\frac{17}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(37\)\(43\)
\( \chi_{ 32718 }(17363, a) \) \(1\)\(1\)\(e\left(\frac{53}{90}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{173}{180}\right)\)\(e\left(\frac{169}{180}\right)\)\(e\left(\frac{79}{90}\right)\)\(e\left(\frac{8}{45}\right)\)\(e\left(\frac{41}{180}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{59}{90}\right)\)
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)
 
\( \chi_{ 32718 }(17363,a) \;\) at \(\;a = \) e.g. 2