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Properties

Label 1305.508
Modulus $1305$
Conductor $1305$
Order $84$
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(1305, base_ring=CyclotomicField(84)) M = H._module chi = DirichletCharacter(H, M([28,63,81]))
 
Copy content gp:[g,chi] = znchar(Mod(508, 1305))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1305.508");
 

Basic properties

Modulus: \(1305\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1305\)
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: 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 1305.db

\(\chi_{1305}(292,\cdot)\) \(\chi_{1305}(322,\cdot)\) \(\chi_{1305}(358,\cdot)\) \(\chi_{1305}(367,\cdot)\) \(\chi_{1305}(403,\cdot)\) \(\chi_{1305}(427,\cdot)\) \(\chi_{1305}(508,\cdot)\) \(\chi_{1305}(562,\cdot)\) \(\chi_{1305}(598,\cdot)\) \(\chi_{1305}(652,\cdot)\) \(\chi_{1305}(727,\cdot)\) \(\chi_{1305}(733,\cdot)\) \(\chi_{1305}(862,\cdot)\) \(\chi_{1305}(868,\cdot)\) \(\chi_{1305}(943,\cdot)\) \(\chi_{1305}(997,\cdot)\) \(\chi_{1305}(1033,\cdot)\) \(\chi_{1305}(1087,\cdot)\) \(\chi_{1305}(1168,\cdot)\) \(\chi_{1305}(1192,\cdot)\) \(\chi_{1305}(1228,\cdot)\) \(\chi_{1305}(1237,\cdot)\) \(\chi_{1305}(1273,\cdot)\) \(\chi_{1305}(1303,\cdot)\)

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

\((146,262,901)\) → \((e\left(\frac{1}{3}\right),-i,e\left(\frac{27}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 1305 }(508, a) \) \(1\)\(1\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{55}{84}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{37}{84}\right)\)\(e\left(\frac{23}{84}\right)\)\(e\left(\frac{59}{84}\right)\)\(e\left(\frac{4}{21}\right)\)\(1\)\(e\left(\frac{5}{28}\right)\)
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)
 
\( \chi_{ 1305 }(508,a) \;\) at \(\;a = \) e.g. 2