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Properties

Label 18901.6
Modulus $18901$
Conductor $18901$
Order $920$
Real no
Primitive yes
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(18901, base_ring=CyclotomicField(920)) M = H._module chi = DirichletCharacter(H, M([23,224]))
 
Copy content gp:[g,chi] = znchar(Mod(6, 18901))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("18901.6");
 

Basic properties

Modulus: \(18901\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(18901\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(920\)
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: yes
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 18901.gw

\(\chi_{18901}(6,\cdot)\) \(\chi_{18901}(67,\cdot)\) \(\chi_{18901}(95,\cdot)\) \(\chi_{18901}(216,\cdot)\) \(\chi_{18901}(227,\cdot)\) \(\chi_{18901}(317,\cdot)\) \(\chi_{18901}(341,\cdot)\) \(\chi_{18901}(358,\cdot)\) \(\chi_{18901}(388,\cdot)\) \(\chi_{18901}(444,\cdot)\) \(\chi_{18901}(457,\cdot)\) \(\chi_{18901}(477,\cdot)\) \(\chi_{18901}(480,\cdot)\) \(\chi_{18901}(504,\cdot)\) \(\chi_{18901}(557,\cdot)\) \(\chi_{18901}(586,\cdot)\) \(\chi_{18901}(598,\cdot)\) \(\chi_{18901}(627,\cdot)\) \(\chi_{18901}(650,\cdot)\) \(\chi_{18901}(685,\cdot)\) \(\chi_{18901}(801,\cdot)\) \(\chi_{18901}(837,\cdot)\) \(\chi_{18901}(958,\cdot)\) \(\chi_{18901}(1060,\cdot)\) \(\chi_{18901}(1113,\cdot)\) \(\chi_{18901}(1120,\cdot)\) \(\chi_{18901}(1176,\cdot)\) \(\chi_{18901}(1241,\cdot)\) \(\chi_{18901}(1278,\cdot)\) \(\chi_{18901}(1334,\cdot)\) ...

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: $\Q(\zeta_{920})$
Fixed field: Number field defined by a degree 920 polynomial (not computed)

Values on generators

\((9682,1846)\) → \((e\left(\frac{1}{40}\right),e\left(\frac{28}{115}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 18901 }(6, a) \) \(-1\)\(1\)\(e\left(\frac{411}{460}\right)\)\(e\left(\frac{369}{920}\right)\)\(e\left(\frac{181}{230}\right)\)\(e\left(\frac{109}{460}\right)\)\(e\left(\frac{271}{920}\right)\)\(e\left(\frac{33}{920}\right)\)\(e\left(\frac{313}{460}\right)\)\(e\left(\frac{369}{460}\right)\)\(e\left(\frac{3}{23}\right)\)\(e\left(\frac{137}{184}\right)\)
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)
 
\( \chi_{ 18901 }(6,a) \;\) at \(\;a = \) e.g. 2