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Properties

Label 3380.167
Modulus $3380$
Conductor $3380$
Order $156$
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(3380, base_ring=CyclotomicField(156)) M = H._module chi = DirichletCharacter(H, M([78,39,79]))
 
Copy content gp:[g,chi] = znchar(Mod(167, 3380))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3380.167");
 

Basic properties

Modulus: \(3380\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3380\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(156\)
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 3380.cs

\(\chi_{3380}(7,\cdot)\) \(\chi_{3380}(123,\cdot)\) \(\chi_{3380}(167,\cdot)\) \(\chi_{3380}(223,\cdot)\) \(\chi_{3380}(267,\cdot)\) \(\chi_{3380}(383,\cdot)\) \(\chi_{3380}(483,\cdot)\) \(\chi_{3380}(527,\cdot)\) \(\chi_{3380}(643,\cdot)\) \(\chi_{3380}(687,\cdot)\) \(\chi_{3380}(743,\cdot)\) \(\chi_{3380}(787,\cdot)\) \(\chi_{3380}(903,\cdot)\) \(\chi_{3380}(947,\cdot)\) \(\chi_{3380}(1003,\cdot)\) \(\chi_{3380}(1047,\cdot)\) \(\chi_{3380}(1163,\cdot)\) \(\chi_{3380}(1207,\cdot)\) \(\chi_{3380}(1307,\cdot)\) \(\chi_{3380}(1423,\cdot)\) \(\chi_{3380}(1467,\cdot)\) \(\chi_{3380}(1523,\cdot)\) \(\chi_{3380}(1567,\cdot)\) \(\chi_{3380}(1683,\cdot)\) \(\chi_{3380}(1727,\cdot)\) \(\chi_{3380}(1783,\cdot)\) \(\chi_{3380}(1827,\cdot)\) \(\chi_{3380}(1943,\cdot)\) \(\chi_{3380}(1987,\cdot)\) \(\chi_{3380}(2043,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((1691,677,1861)\) → \((-1,i,e\left(\frac{79}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 3380 }(167, a) \) \(-1\)\(1\)\(e\left(\frac{7}{156}\right)\)\(e\left(\frac{73}{78}\right)\)\(e\left(\frac{7}{78}\right)\)\(e\left(\frac{103}{156}\right)\)\(e\left(\frac{29}{156}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{51}{52}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{7}{52}\right)\)\(e\left(\frac{59}{78}\right)\)
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)
 
\( \chi_{ 3380 }(167,a) \;\) at \(\;a = \) e.g. 2