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Properties

Label 1472.267
Modulus $1472$
Conductor $1472$
Order $176$
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(1472, base_ring=CyclotomicField(176)) M = H._module chi = DirichletCharacter(H, M([88,55,168]))
 
Copy content gp:[g,chi] = znchar(Mod(267, 1472))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1472.267");
 

Basic properties

Modulus: \(1472\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1472\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(176\)
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 1472.bk

\(\chi_{1472}(11,\cdot)\) \(\chi_{1472}(19,\cdot)\) \(\chi_{1472}(43,\cdot)\) \(\chi_{1472}(51,\cdot)\) \(\chi_{1472}(67,\cdot)\) \(\chi_{1472}(83,\cdot)\) \(\chi_{1472}(99,\cdot)\) \(\chi_{1472}(107,\cdot)\) \(\chi_{1472}(155,\cdot)\) \(\chi_{1472}(171,\cdot)\) \(\chi_{1472}(195,\cdot)\) \(\chi_{1472}(203,\cdot)\) \(\chi_{1472}(227,\cdot)\) \(\chi_{1472}(235,\cdot)\) \(\chi_{1472}(251,\cdot)\) \(\chi_{1472}(267,\cdot)\) \(\chi_{1472}(283,\cdot)\) \(\chi_{1472}(291,\cdot)\) \(\chi_{1472}(339,\cdot)\) \(\chi_{1472}(355,\cdot)\) \(\chi_{1472}(379,\cdot)\) \(\chi_{1472}(387,\cdot)\) \(\chi_{1472}(411,\cdot)\) \(\chi_{1472}(419,\cdot)\) \(\chi_{1472}(435,\cdot)\) \(\chi_{1472}(451,\cdot)\) \(\chi_{1472}(467,\cdot)\) \(\chi_{1472}(475,\cdot)\) \(\chi_{1472}(523,\cdot)\) \(\chi_{1472}(539,\cdot)\) ...

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: $\Q(\zeta_{176})$
Fixed field: Number field defined by a degree 176 polynomial (not computed)

Values on generators

\((1151,645,833)\) → \((-1,e\left(\frac{5}{16}\right),e\left(\frac{21}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1472 }(267, a) \) \(1\)\(1\)\(e\left(\frac{125}{176}\right)\)\(e\left(\frac{47}{176}\right)\)\(e\left(\frac{67}{88}\right)\)\(e\left(\frac{37}{88}\right)\)\(e\left(\frac{115}{176}\right)\)\(e\left(\frac{9}{176}\right)\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{1}{176}\right)\)\(e\left(\frac{83}{176}\right)\)
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)
 
\( \chi_{ 1472 }(267,a) \;\) at \(\;a = \) e.g. 2