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Properties

Label 2484.2305
Modulus $2484$
Conductor $207$
Order $66$
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(2484, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([0,22,3]))
 
Copy content gp:[g,chi] = znchar(Mod(2305, 2484))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2484.2305");
 

Basic properties

Modulus: \(2484\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(207\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(66\)
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 \(\chi_{207}(166,\cdot)\)
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 2484.bh

\(\chi_{2484}(37,\cdot)\) \(\chi_{2484}(145,\cdot)\) \(\chi_{2484}(181,\cdot)\) \(\chi_{2484}(613,\cdot)\) \(\chi_{2484}(793,\cdot)\) \(\chi_{2484}(937,\cdot)\) \(\chi_{2484}(1009,\cdot)\) \(\chi_{2484}(1045,\cdot)\) \(\chi_{2484}(1261,\cdot)\) \(\chi_{2484}(1441,\cdot)\) \(\chi_{2484}(1477,\cdot)\) \(\chi_{2484}(1585,\cdot)\) \(\chi_{2484}(1693,\cdot)\) \(\chi_{2484}(1765,\cdot)\) \(\chi_{2484}(1801,\cdot)\) \(\chi_{2484}(1873,\cdot)\) \(\chi_{2484}(2089,\cdot)\) \(\chi_{2484}(2305,\cdot)\) \(\chi_{2484}(2413,\cdot)\) \(\chi_{2484}(2449,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((1243,461,649)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{1}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 2484 }(2305, a) \) \(-1\)\(1\)\(e\left(\frac{47}{66}\right)\)\(e\left(\frac{13}{66}\right)\)\(e\left(\frac{49}{66}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{10}{11}\right)\)
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)
 
\( \chi_{ 2484 }(2305,a) \;\) at \(\;a = \) e.g. 2