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Properties

Label 2865.578
Modulus $2865$
Conductor $2865$
Order $76$
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(2865, base_ring=CyclotomicField(76)) M = H._module chi = DirichletCharacter(H, M([38,57,20]))
 
Copy content gp:[g,chi] = znchar(Mod(578, 2865))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2865.578");
 

Basic properties

Modulus: \(2865\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2865\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(76\)
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 2865.bj

\(\chi_{2865}(32,\cdot)\) \(\chi_{2865}(107,\cdot)\) \(\chi_{2865}(197,\cdot)\) \(\chi_{2865}(227,\cdot)\) \(\chi_{2865}(368,\cdot)\) \(\chi_{2865}(407,\cdot)\) \(\chi_{2865}(503,\cdot)\) \(\chi_{2865}(518,\cdot)\) \(\chi_{2865}(542,\cdot)\) \(\chi_{2865}(578,\cdot)\) \(\chi_{2865}(698,\cdot)\) \(\chi_{2865}(833,\cdot)\) \(\chi_{2865}(917,\cdot)\) \(\chi_{2865}(1007,\cdot)\) \(\chi_{2865}(1178,\cdot)\) \(\chi_{2865}(1253,\cdot)\) \(\chi_{2865}(1343,\cdot)\) \(\chi_{2865}(1367,\cdot)\) \(\chi_{2865}(1373,\cdot)\) \(\chi_{2865}(1487,\cdot)\) \(\chi_{2865}(1517,\cdot)\) \(\chi_{2865}(1553,\cdot)\) \(\chi_{2865}(1682,\cdot)\) \(\chi_{2865}(1688,\cdot)\) \(\chi_{2865}(2063,\cdot)\) \(\chi_{2865}(2087,\cdot)\) \(\chi_{2865}(2153,\cdot)\) \(\chi_{2865}(2222,\cdot)\) \(\chi_{2865}(2237,\cdot)\) \(\chi_{2865}(2297,\cdot)\) ...

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: $\Q(\zeta_{76})$
Fixed field: Number field defined by a degree 76 polynomial

Values on generators

\((956,1147,2311)\) → \((-1,-i,e\left(\frac{5}{19}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 2865 }(578, a) \) \(1\)\(1\)\(e\left(\frac{63}{76}\right)\)\(e\left(\frac{25}{38}\right)\)\(-i\)\(e\left(\frac{37}{76}\right)\)\(e\left(\frac{33}{38}\right)\)\(e\left(\frac{55}{76}\right)\)\(e\left(\frac{11}{19}\right)\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{3}{76}\right)\)\(e\left(\frac{29}{38}\right)\)
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)
 
\( \chi_{ 2865 }(578,a) \;\) at \(\;a = \) e.g. 2