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Properties

Label 7232.21
Modulus $7232$
Conductor $7232$
Order $112$
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(7232, base_ring=CyclotomicField(112)) M = H._module chi = DirichletCharacter(H, M([0,91,9]))
 
Copy content gp:[g,chi] = znchar(Mod(21, 7232))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7232.21");
 

Basic properties

Modulus: \(7232\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7232\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(112\)
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 7232.gp

\(\chi_{7232}(21,\cdot)\) \(\chi_{7232}(349,\cdot)\) \(\chi_{7232}(373,\cdot)\) \(\chi_{7232}(413,\cdot)\) \(\chi_{7232}(877,\cdot)\) \(\chi_{7232}(909,\cdot)\) \(\chi_{7232}(1333,\cdot)\) \(\chi_{7232}(1389,\cdot)\) \(\chi_{7232}(1549,\cdot)\) \(\chi_{7232}(1629,\cdot)\) \(\chi_{7232}(1909,\cdot)\) \(\chi_{7232}(1989,\cdot)\) \(\chi_{7232}(2029,\cdot)\) \(\chi_{7232}(2037,\cdot)\) \(\chi_{7232}(2061,\cdot)\) \(\chi_{7232}(2077,\cdot)\) \(\chi_{7232}(2109,\cdot)\) \(\chi_{7232}(2277,\cdot)\) \(\chi_{7232}(2653,\cdot)\) \(\chi_{7232}(2709,\cdot)\) \(\chi_{7232}(2837,\cdot)\) \(\chi_{7232}(2957,\cdot)\) \(\chi_{7232}(3045,\cdot)\) \(\chi_{7232}(3109,\cdot)\) \(\chi_{7232}(3301,\cdot)\) \(\chi_{7232}(3413,\cdot)\) \(\chi_{7232}(3901,\cdot)\) \(\chi_{7232}(3909,\cdot)\) \(\chi_{7232}(4373,\cdot)\) \(\chi_{7232}(4445,\cdot)\) ...

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: $\Q(\zeta_{112})$
Fixed field: Number field defined by a degree 112 polynomial (not computed)

Values on generators

\((3391,453,3393)\) → \((1,e\left(\frac{13}{16}\right),e\left(\frac{9}{112}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 7232 }(21, a) \) \(-1\)\(1\)\(e\left(\frac{29}{56}\right)\)\(e\left(\frac{27}{56}\right)\)\(e\left(\frac{43}{56}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{1}{112}\right)\)\(e\left(\frac{107}{112}\right)\)\(1\)\(e\left(\frac{17}{112}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{2}{7}\right)\)
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)
 
\( \chi_{ 7232 }(21,a) \;\) at \(\;a = \) e.g. 2