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Properties

Label 3042.2227
Modulus $3042$
Conductor $1521$
Order $78$
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(3042, base_ring=CyclotomicField(78)) M = H._module chi = DirichletCharacter(H, M([26,67]))
 
Copy content gp:[g,chi] = znchar(Mod(2227, 3042))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3042.2227");
 

Basic properties

Modulus: \(3042\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1521\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(78\)
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 \(\chi_{1521}(706,\cdot)\)
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 3042.br

\(\chi_{3042}(121,\cdot)\) \(\chi_{3042}(205,\cdot)\) \(\chi_{3042}(355,\cdot)\) \(\chi_{3042}(439,\cdot)\) \(\chi_{3042}(589,\cdot)\) \(\chi_{3042}(673,\cdot)\) \(\chi_{3042}(907,\cdot)\) \(\chi_{3042}(1057,\cdot)\) \(\chi_{3042}(1141,\cdot)\) \(\chi_{3042}(1291,\cdot)\) \(\chi_{3042}(1525,\cdot)\) \(\chi_{3042}(1609,\cdot)\) \(\chi_{3042}(1759,\cdot)\) \(\chi_{3042}(1843,\cdot)\) \(\chi_{3042}(1993,\cdot)\) \(\chi_{3042}(2077,\cdot)\) \(\chi_{3042}(2227,\cdot)\) \(\chi_{3042}(2311,\cdot)\) \(\chi_{3042}(2461,\cdot)\) \(\chi_{3042}(2545,\cdot)\) \(\chi_{3042}(2695,\cdot)\) \(\chi_{3042}(2779,\cdot)\) \(\chi_{3042}(2929,\cdot)\) \(\chi_{3042}(3013,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((677,847)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{67}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 3042 }(2227, a) \) \(1\)\(1\)\(e\left(\frac{31}{78}\right)\)\(e\left(\frac{19}{78}\right)\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{16}{39}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{31}{39}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{55}{78}\right)\)\(e\left(\frac{25}{39}\right)\)
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)
 
\( \chi_{ 3042 }(2227,a) \;\) at \(\;a = \) e.g. 2