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Properties

Label 4675.1179
Modulus $4675$
Conductor $4675$
Order $80$
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(4675, base_ring=CyclotomicField(80)) M = H._module chi = DirichletCharacter(H, M([8,8,75]))
 
Copy content gp:[g,chi] = znchar(Mod(1179, 4675))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4675.1179");
 

Basic properties

Modulus: \(4675\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4675\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(80\)
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 4675.ip

\(\chi_{4675}(79,\cdot)\) \(\chi_{4675}(354,\cdot)\) \(\chi_{4675}(369,\cdot)\) \(\chi_{4675}(634,\cdot)\) \(\chi_{4675}(789,\cdot)\) \(\chi_{4675}(904,\cdot)\) \(\chi_{4675}(1064,\cdot)\) \(\chi_{4675}(1179,\cdot)\) \(\chi_{4675}(1184,\cdot)\) \(\chi_{4675}(1459,\cdot)\) \(\chi_{4675}(1469,\cdot)\) \(\chi_{4675}(1729,\cdot)\) \(\chi_{4675}(1744,\cdot)\) \(\chi_{4675}(2009,\cdot)\) \(\chi_{4675}(2164,\cdot)\) \(\chi_{4675}(2284,\cdot)\) \(\chi_{4675}(2714,\cdot)\) \(\chi_{4675}(2829,\cdot)\) \(\chi_{4675}(2834,\cdot)\) \(\chi_{4675}(2844,\cdot)\) \(\chi_{4675}(2989,\cdot)\) \(\chi_{4675}(3104,\cdot)\) \(\chi_{4675}(3394,\cdot)\) \(\chi_{4675}(3539,\cdot)\) \(\chi_{4675}(3669,\cdot)\) \(\chi_{4675}(3814,\cdot)\) \(\chi_{4675}(3934,\cdot)\) \(\chi_{4675}(4204,\cdot)\) \(\chi_{4675}(4209,\cdot)\) \(\chi_{4675}(4219,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((4302,3401,3301)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{1}{10}\right),e\left(\frac{15}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 4675 }(1179, a) \) \(1\)\(1\)\(e\left(\frac{13}{40}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{61}{80}\right)\)\(e\left(\frac{41}{80}\right)\)\(e\left(\frac{39}{40}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{7}{80}\right)\)\(-i\)\(e\left(\frac{67}{80}\right)\)
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)
 
\( \chi_{ 4675 }(1179,a) \;\) at \(\;a = \) e.g. 2