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Properties

Label 35280.17749
Modulus $35280$
Conductor $3920$
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([0,21,0,42,80]))
 
Copy content gp:[g,chi] = znchar(Mod(17749, 35280))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("35280.17749");
 

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: \(3920\)
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 \(\chi_{3920}(2069,\cdot)\)
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.bas

\(\chi_{35280}(109,\cdot)\) \(\chi_{35280}(2629,\cdot)\) \(\chi_{35280}(4069,\cdot)\) \(\chi_{35280}(5149,\cdot)\) \(\chi_{35280}(6589,\cdot)\) \(\chi_{35280}(7669,\cdot)\) \(\chi_{35280}(9109,\cdot)\) \(\chi_{35280}(10189,\cdot)\) \(\chi_{35280}(11629,\cdot)\) \(\chi_{35280}(14149,\cdot)\) \(\chi_{35280}(15229,\cdot)\) \(\chi_{35280}(16669,\cdot)\) \(\chi_{35280}(17749,\cdot)\) \(\chi_{35280}(20269,\cdot)\) \(\chi_{35280}(21709,\cdot)\) \(\chi_{35280}(22789,\cdot)\) \(\chi_{35280}(24229,\cdot)\) \(\chi_{35280}(25309,\cdot)\) \(\chi_{35280}(26749,\cdot)\) \(\chi_{35280}(27829,\cdot)\) \(\chi_{35280}(29269,\cdot)\) \(\chi_{35280}(31789,\cdot)\) \(\chi_{35280}(32869,\cdot)\) \(\chi_{35280}(34309,\cdot)\)

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

\((13231,8821,7841,7057,18721)\) → \((1,i,1,-1,e\left(\frac{20}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 35280 }(17749, a) \) \(1\)\(1\)\(e\left(\frac{29}{84}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{19}{84}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{13}{28}\right)\)
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)
 
\( \chi_{ 35280 }(17749,a) \;\) at \(\;a = \) e.g. 2