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Properties

Label 1045.576
Modulus $1045$
Conductor $209$
Order $45$
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(1045, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([0,18,70]))
 
Copy content gp:[g,chi] = znchar(Mod(576, 1045))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1045.576");
 

Basic properties

Modulus: \(1045\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(209\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(45\)
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 \(\chi_{209}(158,\cdot)\)
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 1045.ce

\(\chi_{1045}(16,\cdot)\) \(\chi_{1045}(36,\cdot)\) \(\chi_{1045}(81,\cdot)\) \(\chi_{1045}(196,\cdot)\) \(\chi_{1045}(251,\cdot)\) \(\chi_{1045}(256,\cdot)\) \(\chi_{1045}(291,\cdot)\) \(\chi_{1045}(301,\cdot)\) \(\chi_{1045}(346,\cdot)\) \(\chi_{1045}(366,\cdot)\) \(\chi_{1045}(416,\cdot)\) \(\chi_{1045}(511,\cdot)\) \(\chi_{1045}(576,\cdot)\) \(\chi_{1045}(586,\cdot)\) \(\chi_{1045}(631,\cdot)\) \(\chi_{1045}(636,\cdot)\) \(\chi_{1045}(731,\cdot)\) \(\chi_{1045}(746,\cdot)\) \(\chi_{1045}(796,\cdot)\) \(\chi_{1045}(841,\cdot)\) \(\chi_{1045}(861,\cdot)\) \(\chi_{1045}(916,\cdot)\) \(\chi_{1045}(966,\cdot)\) \(\chi_{1045}(1016,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: 45.45.43679806300610465846484971330073185012597520004657724600953543350870941304329239684756561.1

Values on generators

\((837,761,496)\) → \((1,e\left(\frac{1}{5}\right),e\left(\frac{7}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(576, a) \) \(1\)\(1\)\(e\left(\frac{44}{45}\right)\)\(e\left(\frac{32}{45}\right)\)\(e\left(\frac{43}{45}\right)\)\(e\left(\frac{31}{45}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{19}{45}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{2}{45}\right)\)
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)
 
\( \chi_{ 1045 }(576,a) \;\) at \(\;a = \) e.g. 2