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Properties

Label 4170.1979
Modulus $4170$
Conductor $2085$
Order $46$
Real no
Primitive no
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(4170, base_ring=CyclotomicField(46)) M = H._module chi = DirichletCharacter(H, M([23,23,39]))
 
Copy content gp:[g,chi] = znchar(Mod(1979, 4170))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4170.1979");
 

Basic properties

Modulus: \(4170\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2085\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(46\)
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: no, induced from \(\chi_{2085}(1979,\cdot)\)
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 4170.bf

\(\chi_{4170}(59,\cdot)\) \(\chi_{4170}(149,\cdot)\) \(\chi_{4170}(479,\cdot)\) \(\chi_{4170}(659,\cdot)\) \(\chi_{4170}(689,\cdot)\) \(\chi_{4170}(779,\cdot)\) \(\chi_{4170}(929,\cdot)\) \(\chi_{4170}(1049,\cdot)\) \(\chi_{4170}(1139,\cdot)\) \(\chi_{4170}(1199,\cdot)\) \(\chi_{4170}(1259,\cdot)\) \(\chi_{4170}(1589,\cdot)\) \(\chi_{4170}(1889,\cdot)\) \(\chi_{4170}(1979,\cdot)\) \(\chi_{4170}(2099,\cdot)\) \(\chi_{4170}(2159,\cdot)\) \(\chi_{4170}(2819,\cdot)\) \(\chi_{4170}(3359,\cdot)\) \(\chi_{4170}(3569,\cdot)\) \(\chi_{4170}(3689,\cdot)\) \(\chi_{4170}(3719,\cdot)\) \(\chi_{4170}(4079,\cdot)\)

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: \(\Q(\zeta_{23})\)
Fixed field: Number field defined by a degree 46 polynomial

Values on generators

\((1391,3337,1531)\) → \((-1,-1,e\left(\frac{39}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 4170 }(1979, a) \) \(1\)\(1\)\(e\left(\frac{41}{46}\right)\)\(e\left(\frac{43}{46}\right)\)\(e\left(\frac{35}{46}\right)\)\(e\left(\frac{33}{46}\right)\)\(e\left(\frac{33}{46}\right)\)\(e\left(\frac{41}{46}\right)\)\(e\left(\frac{9}{46}\right)\)\(e\left(\frac{11}{23}\right)\)\(e\left(\frac{15}{46}\right)\)\(e\left(\frac{29}{46}\right)\)
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)
 
\( \chi_{ 4170 }(1979,a) \;\) at \(\;a = \) e.g. 2