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Properties

Label 388075.10457
Modulus $388075$
Conductor $77615$
Order $1596$
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(388075, base_ring=CyclotomicField(1596)) M = H._module chi = DirichletCharacter(H, M([399,1456,1482]))
 
Copy content gp:[g,chi] = znchar(Mod(10457, 388075))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("388075.10457");
 

Basic properties

Modulus: \(388075\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(77615\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1596\)
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 \(\chi_{77615}(10457,\cdot)\)
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 388075.xo

\(\chi_{388075}(543,\cdot)\) \(\chi_{388075}(2268,\cdot)\) \(\chi_{388075}(3032,\cdot)\) \(\chi_{388075}(3393,\cdot)\) \(\chi_{388075}(4757,\cdot)\) \(\chi_{388075}(4818,\cdot)\) \(\chi_{388075}(5407,\cdot)\) \(\chi_{388075}(6243,\cdot)\) \(\chi_{388075}(7607,\cdot)\) \(\chi_{388075}(9032,\cdot)\) \(\chi_{388075}(9568,\cdot)\) \(\chi_{388075}(10457,\cdot)\) \(\chi_{388075}(11293,\cdot)\) \(\chi_{388075}(11943,\cdot)\) \(\chi_{388075}(13782,\cdot)\) \(\chi_{388075}(14143,\cdot)\) \(\chi_{388075}(14432,\cdot)\) \(\chi_{388075}(15568,\cdot)\) \(\chi_{388075}(16157,\cdot)\) \(\chi_{388075}(16993,\cdot)\) \(\chi_{388075}(17282,\cdot)\) \(\chi_{388075}(18707,\cdot)\) \(\chi_{388075}(20132,\cdot)\) \(\chi_{388075}(20318,\cdot)\) \(\chi_{388075}(20968,\cdot)\) \(\chi_{388075}(22693,\cdot)\) \(\chi_{388075}(23457,\cdot)\) \(\chi_{388075}(23818,\cdot)\) \(\chi_{388075}(25182,\cdot)\) \(\chi_{388075}(25243,\cdot)\) ...

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: $\Q(\zeta_{1596})$
Fixed field: Number field defined by a degree 1596 polynomial (not computed)

Values on generators

\((186277,48376,306851)\) → \((i,e\left(\frac{52}{57}\right),e\left(\frac{13}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 388075 }(10457, a) \) \(1\)\(1\)\(e\left(\frac{373}{1596}\right)\)\(e\left(\frac{775}{1596}\right)\)\(e\left(\frac{373}{798}\right)\)\(e\left(\frac{41}{57}\right)\)\(e\left(\frac{45}{76}\right)\)\(e\left(\frac{373}{532}\right)\)\(e\left(\frac{775}{798}\right)\)\(e\left(\frac{121}{133}\right)\)\(e\left(\frac{507}{532}\right)\)\(e\left(\frac{965}{1596}\right)\)
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)
 
\( \chi_{ 388075 }(10457,a) \;\) at \(\;a = \) e.g. 2