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Properties

Label 3267.2138
Modulus $3267$
Conductor $297$
Order $90$
Real no
Primitive no
Minimal no
Parity odd

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(3267, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([25,18]))
 
Copy content gp:[g,chi] = znchar(Mod(2138, 3267))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3267.2138");
 

Basic properties

Modulus: \(3267\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(297\)
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: no, induced from \(\chi_{297}(59,\cdot)\)
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: no
Parity: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 3267.be

\(\chi_{3267}(245,\cdot)\) \(\chi_{3267}(608,\cdot)\) \(\chi_{3267}(614,\cdot)\) \(\chi_{3267}(632,\cdot)\) \(\chi_{3267}(686,\cdot)\) \(\chi_{3267}(977,\cdot)\) \(\chi_{3267}(995,\cdot)\) \(\chi_{3267}(1049,\cdot)\) \(\chi_{3267}(1334,\cdot)\) \(\chi_{3267}(1697,\cdot)\) \(\chi_{3267}(1703,\cdot)\) \(\chi_{3267}(1721,\cdot)\) \(\chi_{3267}(1775,\cdot)\) \(\chi_{3267}(2066,\cdot)\) \(\chi_{3267}(2084,\cdot)\) \(\chi_{3267}(2138,\cdot)\) \(\chi_{3267}(2423,\cdot)\) \(\chi_{3267}(2786,\cdot)\) \(\chi_{3267}(2792,\cdot)\) \(\chi_{3267}(2810,\cdot)\) \(\chi_{3267}(2864,\cdot)\) \(\chi_{3267}(3155,\cdot)\) \(\chi_{3267}(3173,\cdot)\) \(\chi_{3267}(3227,\cdot)\)

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

\((3026,244)\) → \((e\left(\frac{5}{18}\right),e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 3267 }(2138, a) \) \(-1\)\(1\)\(e\left(\frac{43}{90}\right)\)\(e\left(\frac{43}{45}\right)\)\(e\left(\frac{17}{90}\right)\)\(e\left(\frac{38}{45}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{19}{45}\right)\)\(e\left(\frac{29}{90}\right)\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{29}{30}\right)\)
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)
 
\( \chi_{ 3267 }(2138,a) \;\) at \(\;a = \) e.g. 2