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Properties

Label 11385.1183
Modulus $11385$
Conductor $11385$
Order $660$
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(11385, base_ring=CyclotomicField(660)) M = H._module chi = DirichletCharacter(H, M([220,495,594,90]))
 
Copy content gp:[g,chi] = znchar(Mod(1183, 11385))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11385.1183");
 

Basic properties

Modulus: \(11385\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(11385\)
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: 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 11385.hd

\(\chi_{11385}(7,\cdot)\) \(\chi_{11385}(112,\cdot)\) \(\chi_{11385}(178,\cdot)\) \(\chi_{11385}(283,\cdot)\) \(\chi_{11385}(337,\cdot)\) \(\chi_{11385}(382,\cdot)\) \(\chi_{11385}(448,\cdot)\) \(\chi_{11385}(457,\cdot)\) \(\chi_{11385}(502,\cdot)\) \(\chi_{11385}(688,\cdot)\) \(\chi_{11385}(733,\cdot)\) \(\chi_{11385}(778,\cdot)\) \(\chi_{11385}(787,\cdot)\) \(\chi_{11385}(1003,\cdot)\) \(\chi_{11385}(1042,\cdot)\) \(\chi_{11385}(1102,\cdot)\) \(\chi_{11385}(1183,\cdot)\) \(\chi_{11385}(1282,\cdot)\) \(\chi_{11385}(1348,\cdot)\) \(\chi_{11385}(1372,\cdot)\) \(\chi_{11385}(1447,\cdot)\) \(\chi_{11385}(1492,\cdot)\) \(\chi_{11385}(1537,\cdot)\)

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

\((2531,6832,1036,3961)\) → \((e\left(\frac{1}{3}\right),-i,e\left(\frac{9}{10}\right),e\left(\frac{3}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(26\)
\( \chi_{ 11385 }(1183, a) \) \(-1\)\(1\)\(e\left(\frac{169}{660}\right)\)\(e\left(\frac{169}{330}\right)\)\(e\left(\frac{643}{660}\right)\)\(e\left(\frac{169}{220}\right)\)\(e\left(\frac{479}{660}\right)\)\(e\left(\frac{38}{165}\right)\)\(e\left(\frac{4}{165}\right)\)\(e\left(\frac{177}{220}\right)\)\(e\left(\frac{27}{110}\right)\)\(e\left(\frac{54}{55}\right)\)
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)
 
\( \chi_{ 11385 }(1183,a) \;\) at \(\;a = \) e.g. 2