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Properties

Label 1309.260
Modulus $1309$
Conductor $187$
Order $80$
Real no
Primitive no
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(1309, base_ring=CyclotomicField(80)) M = H._module chi = DirichletCharacter(H, M([0,56,25]))
 
Copy content gp:[g,chi] = znchar(Mod(260, 1309))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1309.260");
 

Basic properties

Modulus: \(1309\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(187\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(80\)
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: no, induced from \(\chi_{187}(73,\cdot)\)
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 1309.ct

\(\chi_{1309}(29,\cdot)\) \(\chi_{1309}(57,\cdot)\) \(\chi_{1309}(211,\cdot)\) \(\chi_{1309}(260,\cdot)\) \(\chi_{1309}(316,\cdot)\) \(\chi_{1309}(337,\cdot)\) \(\chi_{1309}(414,\cdot)\) \(\chi_{1309}(435,\cdot)\) \(\chi_{1309}(470,\cdot)\) \(\chi_{1309}(547,\cdot)\) \(\chi_{1309}(568,\cdot)\) \(\chi_{1309}(589,\cdot)\) \(\chi_{1309}(624,\cdot)\) \(\chi_{1309}(666,\cdot)\) \(\chi_{1309}(673,\cdot)\) \(\chi_{1309}(743,\cdot)\) \(\chi_{1309}(827,\cdot)\) \(\chi_{1309}(855,\cdot)\) \(\chi_{1309}(904,\cdot)\) \(\chi_{1309}(932,\cdot)\) \(\chi_{1309}(974,\cdot)\) \(\chi_{1309}(981,\cdot)\) \(\chi_{1309}(1009,\cdot)\) \(\chi_{1309}(1030,\cdot)\) \(\chi_{1309}(1051,\cdot)\) \(\chi_{1309}(1128,\cdot)\) \(\chi_{1309}(1163,\cdot)\) \(\chi_{1309}(1184,\cdot)\) \(\chi_{1309}(1212,\cdot)\) \(\chi_{1309}(1261,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((1123,596,309)\) → \((1,e\left(\frac{7}{10}\right),e\left(\frac{5}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 1309 }(260, a) \) \(1\)\(1\)\(e\left(\frac{3}{40}\right)\)\(e\left(\frac{73}{80}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{29}{80}\right)\)\(e\left(\frac{79}{80}\right)\)\(e\left(\frac{9}{40}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{19}{20}\right)\)
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)
 
\( \chi_{ 1309 }(260,a) \;\) at \(\;a = \) e.g. 2