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Properties

Label 1573.291
Modulus $1573$
Conductor $1573$
Order $220$
Real no
Primitive yes
Minimal yes
Parity odd

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(1573, base_ring=CyclotomicField(220)) M = H._module chi = DirichletCharacter(H, M([28,165]))
 
Copy content gp:[g,chi] = znchar(Mod(291, 1573))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1573.291");
 

Basic properties

Modulus: \(1573\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1573\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(220\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1573.bq

\(\chi_{1573}(5,\cdot)\) \(\chi_{1573}(31,\cdot)\) \(\chi_{1573}(47,\cdot)\) \(\chi_{1573}(60,\cdot)\) \(\chi_{1573}(70,\cdot)\) \(\chi_{1573}(86,\cdot)\) \(\chi_{1573}(125,\cdot)\) \(\chi_{1573}(135,\cdot)\) \(\chi_{1573}(174,\cdot)\) \(\chi_{1573}(190,\cdot)\) \(\chi_{1573}(203,\cdot)\) \(\chi_{1573}(213,\cdot)\) \(\chi_{1573}(229,\cdot)\) \(\chi_{1573}(268,\cdot)\) \(\chi_{1573}(278,\cdot)\) \(\chi_{1573}(291,\cdot)\) \(\chi_{1573}(317,\cdot)\) \(\chi_{1573}(333,\cdot)\) \(\chi_{1573}(346,\cdot)\) \(\chi_{1573}(356,\cdot)\) \(\chi_{1573}(411,\cdot)\) \(\chi_{1573}(421,\cdot)\) \(\chi_{1573}(434,\cdot)\) \(\chi_{1573}(460,\cdot)\) \(\chi_{1573}(476,\cdot)\) \(\chi_{1573}(489,\cdot)\) \(\chi_{1573}(499,\cdot)\) \(\chi_{1573}(515,\cdot)\) \(\chi_{1573}(554,\cdot)\) \(\chi_{1573}(564,\cdot)\) ...

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: $\Q(\zeta_{220})$
Fixed field: Number field defined by a degree 220 polynomial (not computed)

Values on generators

\((365,1211)\) → \((e\left(\frac{7}{55}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1573 }(291, a) \) \(-1\)\(1\)\(e\left(\frac{193}{220}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{83}{110}\right)\)\(e\left(\frac{37}{220}\right)\)\(e\left(\frac{17}{220}\right)\)\(e\left(\frac{31}{220}\right)\)\(e\left(\frac{139}{220}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{21}{22}\right)\)
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)
 
\( \chi_{ 1573 }(291,a) \;\) at \(\;a = \) e.g. 2