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Properties

Label 2279.40
Modulus $2279$
Conductor $2279$
Order $546$
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(2279, base_ring=CyclotomicField(546)) M = H._module chi = DirichletCharacter(H, M([286,525]))
 
Copy content gp:[g,chi] = znchar(Mod(40, 2279))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2279.40");
 

Basic properties

Modulus: \(2279\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2279\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(546\)
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 2279.bs

\(\chi_{2279}(9,\cdot)\) \(\chi_{2279}(17,\cdot)\) \(\chi_{2279}(25,\cdot)\) \(\chi_{2279}(38,\cdot)\) \(\chi_{2279}(40,\cdot)\) \(\chi_{2279}(57,\cdot)\) \(\chi_{2279}(60,\cdot)\) \(\chi_{2279}(96,\cdot)\) \(\chi_{2279}(110,\cdot)\) \(\chi_{2279}(117,\cdot)\) \(\chi_{2279}(143,\cdot)\) \(\chi_{2279}(144,\cdot)\) \(\chi_{2279}(146,\cdot)\) \(\chi_{2279}(196,\cdot)\) \(\chi_{2279}(197,\cdot)\) \(\chi_{2279}(229,\cdot)\) \(\chi_{2279}(255,\cdot)\) \(\chi_{2279}(271,\cdot)\) \(\chi_{2279}(272,\cdot)\) \(\chi_{2279}(282,\cdot)\) \(\chi_{2279}(324,\cdot)\) \(\chi_{2279}(325,\cdot)\) \(\chi_{2279}(358,\cdot)\) \(\chi_{2279}(361,\cdot)\) \(\chi_{2279}(375,\cdot)\) \(\chi_{2279}(382,\cdot)\) \(\chi_{2279}(396,\cdot)\) \(\chi_{2279}(400,\cdot)\) \(\chi_{2279}(411,\cdot)\) \(\chi_{2279}(453,\cdot)\) ...

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: $\Q(\zeta_{273})$
Fixed field: Number field defined by a degree 546 polynomial (not computed)

Values on generators

\((1379,1592)\) → \((e\left(\frac{11}{21}\right),e\left(\frac{25}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2279 }(40, a) \) \(1\)\(1\)\(e\left(\frac{19}{182}\right)\)\(e\left(\frac{475}{546}\right)\)\(e\left(\frac{19}{91}\right)\)\(e\left(\frac{157}{546}\right)\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{31}{39}\right)\)\(e\left(\frac{57}{182}\right)\)\(e\left(\frac{202}{273}\right)\)\(e\left(\frac{107}{273}\right)\)\(e\left(\frac{44}{91}\right)\)
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)
 
\( \chi_{ 2279 }(40,a) \;\) at \(\;a = \) e.g. 2