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Properties

Label 2575.1091
Modulus $2575$
Conductor $2575$
Order $85$
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(2575, base_ring=CyclotomicField(170)) M = H._module chi = DirichletCharacter(H, M([34,60]))
 
Copy content gp:[g,chi] = znchar(Mod(1091, 2575))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2575.1091");
 

Basic properties

Modulus: \(2575\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2575\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(85\)
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 2575.bf

\(\chi_{2575}(61,\cdot)\) \(\chi_{2575}(66,\cdot)\) \(\chi_{2575}(81,\cdot)\) \(\chi_{2575}(111,\cdot)\) \(\chi_{2575}(116,\cdot)\) \(\chi_{2575}(196,\cdot)\) \(\chi_{2575}(236,\cdot)\) \(\chi_{2575}(306,\cdot)\) \(\chi_{2575}(381,\cdot)\) \(\chi_{2575}(421,\cdot)\) \(\chi_{2575}(446,\cdot)\) \(\chi_{2575}(491,\cdot)\) \(\chi_{2575}(581,\cdot)\) \(\chi_{2575}(591,\cdot)\) \(\chi_{2575}(596,\cdot)\) \(\chi_{2575}(631,\cdot)\) \(\chi_{2575}(641,\cdot)\) \(\chi_{2575}(711,\cdot)\) \(\chi_{2575}(821,\cdot)\) \(\chi_{2575}(896,\cdot)\) \(\chi_{2575}(936,\cdot)\) \(\chi_{2575}(941,\cdot)\) \(\chi_{2575}(961,\cdot)\) \(\chi_{2575}(991,\cdot)\) \(\chi_{2575}(1006,\cdot)\) \(\chi_{2575}(1091,\cdot)\) \(\chi_{2575}(1096,\cdot)\) \(\chi_{2575}(1106,\cdot)\) \(\chi_{2575}(1111,\cdot)\) \(\chi_{2575}(1141,\cdot)\) ...

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: $\Q(\zeta_{85})$
Fixed field: Number field defined by a degree 85 polynomial

Values on generators

\((1752,726)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{6}{17}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 2575 }(1091, a) \) \(1\)\(1\)\(e\left(\frac{62}{85}\right)\)\(e\left(\frac{14}{85}\right)\)\(e\left(\frac{39}{85}\right)\)\(e\left(\frac{76}{85}\right)\)\(e\left(\frac{7}{17}\right)\)\(e\left(\frac{16}{85}\right)\)\(e\left(\frac{28}{85}\right)\)\(e\left(\frac{62}{85}\right)\)\(e\left(\frac{53}{85}\right)\)\(e\left(\frac{18}{85}\right)\)
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)
 
\( \chi_{ 2575 }(1091,a) \;\) at \(\;a = \) e.g. 2