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Properties

Label 32200.1823
Modulus $32200$
Conductor $16100$
Order $660$
Real no
Primitive no
Minimal no
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(32200, base_ring=CyclotomicField(660)) M = H._module chi = DirichletCharacter(H, M([330,0,363,110,540]))
 
Copy content gp:[g,chi] = znchar(Mod(1823, 32200))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("32200.1823");
 

Basic properties

Modulus: \(32200\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(16100\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(660\)
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 \(\chi_{16100}(1823,\cdot)\)
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 32200.oq

\(\chi_{32200}(87,\cdot)\) \(\chi_{32200}(423,\cdot)\) \(\chi_{32200}(647,\cdot)\) \(\chi_{32200}(703,\cdot)\) \(\chi_{32200}(887,\cdot)\) \(\chi_{32200}(1223,\cdot)\) \(\chi_{32200}(1503,\cdot)\) \(\chi_{32200}(1727,\cdot)\) \(\chi_{32200}(1783,\cdot)\) \(\chi_{32200}(1823,\cdot)\) \(\chi_{32200}(2063,\cdot)\) \(\chi_{32200}(2327,\cdot)\) \(\chi_{32200}(2663,\cdot)\) \(\chi_{32200}(2847,\cdot)\) \(\chi_{32200}(2887,\cdot)\) \(\chi_{32200}(3167,\cdot)\) \(\chi_{32200}(3183,\cdot)\) \(\chi_{32200}(3223,\cdot)\) \(\chi_{32200}(3463,\cdot)\) \(\chi_{32200}(4287,\cdot)\) \(\chi_{32200}(4303,\cdot)\) \(\chi_{32200}(4567,\cdot)\) \(\chi_{32200}(4583,\cdot)\) \(\chi_{32200}(4903,\cdot)\) \(\chi_{32200}(5087,\cdot)\) \(\chi_{32200}(5183,\cdot)\) \(\chi_{32200}(5367,\cdot)\) \(\chi_{32200}(5423,\cdot)\) \(\chi_{32200}(5463,\cdot)\) \(\chi_{32200}(5647,\cdot)\) ...

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: $\Q(\zeta_{660})$
Fixed field: Number field defined by a degree 660 polynomial (not computed)

Values on generators

\((24151,16101,2577,9201,19601)\) → \((-1,1,e\left(\frac{11}{20}\right),e\left(\frac{1}{6}\right),e\left(\frac{9}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(27\)\(29\)\(31\)\(33\)
\( \chi_{ 32200 }(1823, a) \) \(-1\)\(1\)\(e\left(\frac{401}{660}\right)\)\(e\left(\frac{71}{330}\right)\)\(e\left(\frac{109}{330}\right)\)\(e\left(\frac{89}{220}\right)\)\(e\left(\frac{29}{660}\right)\)\(e\left(\frac{167}{330}\right)\)\(e\left(\frac{181}{220}\right)\)\(e\left(\frac{91}{110}\right)\)\(e\left(\frac{161}{165}\right)\)\(e\left(\frac{619}{660}\right)\)
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)
 
\( \chi_{ 32200 }(1823,a) \;\) at \(\;a = \) e.g. 2