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Properties

Label 7360.757
Modulus $7360$
Conductor $7360$
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(7360, base_ring=CyclotomicField(176)) M = H._module chi = DirichletCharacter(H, M([0,55,44,104]))
 
Copy content gp:[g,chi] = znchar(Mod(757, 7360))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7360.757");
 

Basic properties

Modulus: \(7360\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7360\)
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 7360.fn

\(\chi_{7360}(37,\cdot)\) \(\chi_{7360}(333,\cdot)\) \(\chi_{7360}(493,\cdot)\) \(\chi_{7360}(517,\cdot)\) \(\chi_{7360}(573,\cdot)\) \(\chi_{7360}(677,\cdot)\) \(\chi_{7360}(733,\cdot)\) \(\chi_{7360}(757,\cdot)\) \(\chi_{7360}(893,\cdot)\) \(\chi_{7360}(917,\cdot)\) \(\chi_{7360}(973,\cdot)\) \(\chi_{7360}(1077,\cdot)\) \(\chi_{7360}(1157,\cdot)\) \(\chi_{7360}(1213,\cdot)\) \(\chi_{7360}(1293,\cdot)\) \(\chi_{7360}(1397,\cdot)\) \(\chi_{7360}(1477,\cdot)\) \(\chi_{7360}(1533,\cdot)\) \(\chi_{7360}(1693,\cdot)\) \(\chi_{7360}(1717,\cdot)\) \(\chi_{7360}(1877,\cdot)\) \(\chi_{7360}(2173,\cdot)\) \(\chi_{7360}(2333,\cdot)\) \(\chi_{7360}(2357,\cdot)\) \(\chi_{7360}(2413,\cdot)\) \(\chi_{7360}(2517,\cdot)\) \(\chi_{7360}(2573,\cdot)\) \(\chi_{7360}(2597,\cdot)\) \(\chi_{7360}(2733,\cdot)\) \(\chi_{7360}(2757,\cdot)\) ...

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

\((1151,5061,4417,6721)\) → \((1,e\left(\frac{5}{16}\right),i,e\left(\frac{13}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 7360 }(757, a) \) \(1\)\(1\)\(e\left(\frac{25}{176}\right)\)\(e\left(\frac{53}{88}\right)\)\(e\left(\frac{25}{88}\right)\)\(e\left(\frac{155}{176}\right)\)\(e\left(\frac{125}{176}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{97}{176}\right)\)\(e\left(\frac{131}{176}\right)\)\(e\left(\frac{75}{176}\right)\)\(e\left(\frac{101}{176}\right)\)
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)
 
\( \chi_{ 7360 }(757,a) \;\) at \(\;a = \) e.g. 2