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Properties

Label 3968.685
Modulus $3968$
Conductor $3968$
Order $480$
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(3968, base_ring=CyclotomicField(480)) M = H._module chi = DirichletCharacter(H, M([0,105,16]))
 
Copy content gp:[g,chi] = znchar(Mod(685, 3968))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3968.685");
 

Basic properties

Modulus: \(3968\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3968\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(480\)
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 3968.dp

\(\chi_{3968}(13,\cdot)\) \(\chi_{3968}(21,\cdot)\) \(\chi_{3968}(53,\cdot)\) \(\chi_{3968}(117,\cdot)\) \(\chi_{3968}(141,\cdot)\) \(\chi_{3968}(189,\cdot)\) \(\chi_{3968}(197,\cdot)\) \(\chi_{3968}(229,\cdot)\) \(\chi_{3968}(261,\cdot)\) \(\chi_{3968}(269,\cdot)\) \(\chi_{3968}(301,\cdot)\) \(\chi_{3968}(365,\cdot)\) \(\chi_{3968}(389,\cdot)\) \(\chi_{3968}(437,\cdot)\) \(\chi_{3968}(445,\cdot)\) \(\chi_{3968}(477,\cdot)\) \(\chi_{3968}(509,\cdot)\) \(\chi_{3968}(517,\cdot)\) \(\chi_{3968}(549,\cdot)\) \(\chi_{3968}(613,\cdot)\) \(\chi_{3968}(637,\cdot)\) \(\chi_{3968}(685,\cdot)\) \(\chi_{3968}(693,\cdot)\) \(\chi_{3968}(725,\cdot)\) \(\chi_{3968}(757,\cdot)\) \(\chi_{3968}(765,\cdot)\) \(\chi_{3968}(797,\cdot)\) \(\chi_{3968}(861,\cdot)\) \(\chi_{3968}(885,\cdot)\) \(\chi_{3968}(933,\cdot)\) ...

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: $\Q(\zeta_{480})$
Fixed field: Number field defined by a degree 480 polynomial (not computed)

Values on generators

\((2047,3845,2049)\) → \((1,e\left(\frac{7}{32}\right),e\left(\frac{1}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 3968 }(685, a) \) \(-1\)\(1\)\(e\left(\frac{331}{480}\right)\)\(e\left(\frac{85}{96}\right)\)\(e\left(\frac{29}{240}\right)\)\(e\left(\frac{91}{240}\right)\)\(e\left(\frac{173}{480}\right)\)\(e\left(\frac{311}{480}\right)\)\(e\left(\frac{23}{40}\right)\)\(e\left(\frac{43}{120}\right)\)\(e\left(\frac{79}{480}\right)\)\(e\left(\frac{389}{480}\right)\)
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)
 
\( \chi_{ 3968 }(685,a) \;\) at \(\;a = \) e.g. 2