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Properties

Label 119427.26
Modulus $119427$
Conductor $119427$
Order $7590$
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(119427, base_ring=CyclotomicField(7590)) M = H._module chi = DirichletCharacter(H, M([3795,6325,7038,4785]))
 
Copy content gp:[g,chi] = znchar(Mod(26, 119427))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("119427.26");
 

Basic properties

Modulus: \(119427\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(119427\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(7590\)
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 119427.jj

\(\chi_{119427}(5,\cdot)\) \(\chi_{119427}(26,\cdot)\) \(\chi_{119427}(38,\cdot)\) \(\chi_{119427}(80,\cdot)\) \(\chi_{119427}(152,\cdot)\) \(\chi_{119427}(185,\cdot)\) \(\chi_{119427}(257,\cdot)\) \(\chi_{119427}(278,\cdot)\) \(\chi_{119427}(311,\cdot)\) \(\chi_{119427}(416,\cdot)\) \(\chi_{119427}(467,\cdot)\) \(\chi_{119427}(500,\cdot)\) \(\chi_{119427}(509,\cdot)\) \(\chi_{119427}(698,\cdot)\) \(\chi_{119427}(731,\cdot)\) \(\chi_{119427}(740,\cdot)\) \(\chi_{119427}(950,\cdot)\) \(\chi_{119427}(962,\cdot)\) \(\chi_{119427}(983,\cdot)\) \(\chi_{119427}(1214,\cdot)\) \(\chi_{119427}(1235,\cdot)\) \(\chi_{119427}(1307,\cdot)\) \(\chi_{119427}(1433,\cdot)\) \(\chi_{119427}(1445,\cdot)\)

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: $\Q(\zeta_{3795})$
Fixed field: Number field defined by a degree 7590 polynomial (not computed)

Values on generators

\((79619,17062,50338,43198)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{51}{55}\right),e\left(\frac{29}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 119427 }(26, a) \) \(-1\)\(1\)\(e\left(\frac{3353}{7590}\right)\)\(e\left(\frac{3353}{3795}\right)\)\(e\left(\frac{6947}{7590}\right)\)\(e\left(\frac{823}{2530}\right)\)\(e\left(\frac{271}{759}\right)\)\(e\left(\frac{113}{1265}\right)\)\(e\left(\frac{2911}{3795}\right)\)\(e\left(\frac{3251}{3795}\right)\)\(e\left(\frac{1897}{3795}\right)\)\(e\left(\frac{2021}{2530}\right)\)
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)
 
\( \chi_{ 119427 }(26,a) \;\) at \(\;a = \) e.g. 2