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Properties

Label 26257.4980
Modulus $26257$
Conductor $26257$
Order $330$
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(26257, base_ring=CyclotomicField(330)) M = H._module chi = DirichletCharacter(H, M([55,249,88]))
 
Copy content gp:[g,chi] = znchar(Mod(4980, 26257))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("26257.4980");
 

Basic properties

Modulus: \(26257\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(26257\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(330\)
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 26257.ui

\(\chi_{26257}(206,\cdot)\) \(\chi_{26257}(381,\cdot)\) \(\chi_{26257}(948,\cdot)\) \(\chi_{26257}(1095,\cdot)\) \(\chi_{26257}(1454,\cdot)\) \(\chi_{26257}(1657,\cdot)\) \(\chi_{26257}(1755,\cdot)\) \(\chi_{26257}(1867,\cdot)\) \(\chi_{26257}(2593,\cdot)\) \(\chi_{26257}(2768,\cdot)\) \(\chi_{26257}(3335,\cdot)\) \(\chi_{26257}(3841,\cdot)\) \(\chi_{26257}(4044,\cdot)\) \(\chi_{26257}(4142,\cdot)\) \(\chi_{26257}(4254,\cdot)\) \(\chi_{26257}(4980,\cdot)\) \(\chi_{26257}(5155,\cdot)\) \(\chi_{26257}(5722,\cdot)\) \(\chi_{26257}(5869,\cdot)\) \(\chi_{26257}(6228,\cdot)\) \(\chi_{26257}(6431,\cdot)\) \(\chi_{26257}(6529,\cdot)\) \(\chi_{26257}(6641,\cdot)\) \(\chi_{26257}(7367,\cdot)\) \(\chi_{26257}(8109,\cdot)\) \(\chi_{26257}(8256,\cdot)\) \(\chi_{26257}(8615,\cdot)\) \(\chi_{26257}(8818,\cdot)\) \(\chi_{26257}(8916,\cdot)\) \(\chi_{26257}(9028,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((18756,6294,21176)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{83}{110}\right),e\left(\frac{4}{15}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 26257 }(4980, a) \) \(1\)\(1\)\(e\left(\frac{161}{330}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{161}{165}\right)\)\(e\left(\frac{1}{330}\right)\)\(e\left(\frac{53}{165}\right)\)\(e\left(\frac{51}{110}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{27}{55}\right)\)\(e\left(\frac{89}{110}\right)\)\(e\left(\frac{106}{165}\right)\)
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)
 
\( \chi_{ 26257 }(4980,a) \;\) at \(\;a = \) e.g. 2