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Properties

Label 35280.19171
Modulus $35280$
Conductor $784$
Order $84$
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(35280, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([42,63,0,0,22]))
 
Copy content gp:[g,chi] = znchar(Mod(19171, 35280))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("35280.19171");
 

Basic properties

Modulus: \(35280\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(784\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(84\)
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 \(\chi_{784}(355,\cdot)\)
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 35280.bar

\(\chi_{35280}(451,\cdot)\) \(\chi_{35280}(1531,\cdot)\) \(\chi_{35280}(4051,\cdot)\) \(\chi_{35280}(5491,\cdot)\) \(\chi_{35280}(6571,\cdot)\) \(\chi_{35280}(8011,\cdot)\) \(\chi_{35280}(9091,\cdot)\) \(\chi_{35280}(10531,\cdot)\) \(\chi_{35280}(11611,\cdot)\) \(\chi_{35280}(13051,\cdot)\) \(\chi_{35280}(15571,\cdot)\) \(\chi_{35280}(16651,\cdot)\) \(\chi_{35280}(18091,\cdot)\) \(\chi_{35280}(19171,\cdot)\) \(\chi_{35280}(21691,\cdot)\) \(\chi_{35280}(23131,\cdot)\) \(\chi_{35280}(24211,\cdot)\) \(\chi_{35280}(25651,\cdot)\) \(\chi_{35280}(26731,\cdot)\) \(\chi_{35280}(28171,\cdot)\) \(\chi_{35280}(29251,\cdot)\) \(\chi_{35280}(30691,\cdot)\) \(\chi_{35280}(33211,\cdot)\) \(\chi_{35280}(34291,\cdot)\)

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: $\Q(\zeta_{84})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 84 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((13231,8821,7841,7057,18721)\) → \((-1,-i,1,1,e\left(\frac{11}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 35280 }(19171, a) \) \(1\)\(1\)\(e\left(\frac{61}{84}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{84}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{23}{28}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 35280 }(19171,a) \;\) at \(\;a = \) e.g. 2