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Properties

Label 6105.1637
Modulus $6105$
Conductor $6105$
Order $180$
Real no
Primitive yes
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(6105, base_ring=CyclotomicField(180)) M = H._module chi = DirichletCharacter(H, M([90,45,108,80]))
 
Copy content gp:[g,chi] = znchar(Mod(1637, 6105))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6105.1637");
 

Basic properties

Modulus: \(6105\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(6105\)
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: 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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 6105.iz

\(\chi_{6105}(53,\cdot)\) \(\chi_{6105}(218,\cdot)\) \(\chi_{6105}(377,\cdot)\) \(\chi_{6105}(488,\cdot)\) \(\chi_{6105}(608,\cdot)\) \(\chi_{6105}(752,\cdot)\) \(\chi_{6105}(773,\cdot)\) \(\chi_{6105}(863,\cdot)\) \(\chi_{6105}(1043,\cdot)\) \(\chi_{6105}(1082,\cdot)\) \(\chi_{6105}(1193,\cdot)\) \(\chi_{6105}(1292,\cdot)\) \(\chi_{6105}(1307,\cdot)\) \(\chi_{6105}(1598,\cdot)\) \(\chi_{6105}(1637,\cdot)\) \(\chi_{6105}(1862,\cdot)\) \(\chi_{6105}(1973,\cdot)\) \(\chi_{6105}(2192,\cdot)\) \(\chi_{6105}(2303,\cdot)\) \(\chi_{6105}(2402,\cdot)\) \(\chi_{6105}(2513,\cdot)\) \(\chi_{6105}(2528,\cdot)\) \(\chi_{6105}(2858,\cdot)\) \(\chi_{6105}(2957,\cdot)\) \(\chi_{6105}(3083,\cdot)\) \(\chi_{6105}(3272,\cdot)\) \(\chi_{6105}(3413,\cdot)\) \(\chi_{6105}(3437,\cdot)\) \(\chi_{6105}(3512,\cdot)\) \(\chi_{6105}(3623,\cdot)\) ...

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

\((2036,1222,4996,3961)\) → \((-1,i,e\left(\frac{3}{5}\right),e\left(\frac{4}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(23\)
\( \chi_{ 6105 }(1637, a) \) \(1\)\(1\)\(e\left(\frac{143}{180}\right)\)\(e\left(\frac{53}{90}\right)\)\(e\left(\frac{121}{180}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{43}{180}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{8}{45}\right)\)\(e\left(\frac{47}{180}\right)\)\(e\left(\frac{77}{90}\right)\)\(e\left(\frac{11}{12}\right)\)
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)
 
\( \chi_{ 6105 }(1637,a) \;\) at \(\;a = \) e.g. 2