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Properties

Label 3861.3085
Modulus $3861$
Conductor $3861$
Order $90$
Real no
Primitive yes
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(3861, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([80,36,15]))
 
Copy content gp:[g,chi] = znchar(Mod(3085, 3861))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3861.3085");
 

Basic properties

Modulus: \(3861\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3861\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(90\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 3861.go

\(\chi_{3861}(49,\cdot)\) \(\chi_{3861}(394,\cdot)\) \(\chi_{3861}(400,\cdot)\) \(\chi_{3861}(511,\cdot)\) \(\chi_{3861}(751,\cdot)\) \(\chi_{3861}(862,\cdot)\) \(\chi_{3861}(1213,\cdot)\) \(\chi_{3861}(1219,\cdot)\) \(\chi_{3861}(1336,\cdot)\) \(\chi_{3861}(1681,\cdot)\) \(\chi_{3861}(1687,\cdot)\) \(\chi_{3861}(1798,\cdot)\) \(\chi_{3861}(2038,\cdot)\) \(\chi_{3861}(2149,\cdot)\) \(\chi_{3861}(2500,\cdot)\) \(\chi_{3861}(2506,\cdot)\) \(\chi_{3861}(2623,\cdot)\) \(\chi_{3861}(2968,\cdot)\) \(\chi_{3861}(2974,\cdot)\) \(\chi_{3861}(3085,\cdot)\) \(\chi_{3861}(3325,\cdot)\) \(\chi_{3861}(3436,\cdot)\) \(\chi_{3861}(3787,\cdot)\) \(\chi_{3861}(3793,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((2432,3511,1783)\) → \((e\left(\frac{8}{9}\right),e\left(\frac{2}{5}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 3861 }(3085, a) \) \(1\)\(1\)\(e\left(\frac{41}{90}\right)\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{49}{90}\right)\)\(e\left(\frac{77}{90}\right)\)\(e\left(\frac{11}{30}\right)\)\(1\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{37}{45}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{7}{10}\right)\)
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)
 
\( \chi_{ 3861 }(3085,a) \;\) at \(\;a = \) e.g. 2