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Properties

Label 9408.5063
Modulus $9408$
Conductor $4704$
Order $168$
Real no
Primitive no
Minimal no
Parity even

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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(9408, base_ring=CyclotomicField(168)) M = H._module chi = DirichletCharacter(H, M([84,105,84,80]))
 
Copy content gp:[g,chi] = znchar(Mod(5063, 9408))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9408.5063");
 

Basic properties

Modulus: \(9408\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4704\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(168\)
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 \(\chi_{4704}(2123,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 9408.fu

\(\chi_{9408}(23,\cdot)\) \(\chi_{9408}(359,\cdot)\) \(\chi_{9408}(599,\cdot)\) \(\chi_{9408}(695,\cdot)\) \(\chi_{9408}(935,\cdot)\) \(\chi_{9408}(1031,\cdot)\) \(\chi_{9408}(1271,\cdot)\) \(\chi_{9408}(1367,\cdot)\) \(\chi_{9408}(1607,\cdot)\) \(\chi_{9408}(1703,\cdot)\) \(\chi_{9408}(1943,\cdot)\) \(\chi_{9408}(2279,\cdot)\) \(\chi_{9408}(2375,\cdot)\) \(\chi_{9408}(2711,\cdot)\) \(\chi_{9408}(2951,\cdot)\) \(\chi_{9408}(3047,\cdot)\) \(\chi_{9408}(3287,\cdot)\) \(\chi_{9408}(3383,\cdot)\) \(\chi_{9408}(3623,\cdot)\) \(\chi_{9408}(3719,\cdot)\) \(\chi_{9408}(3959,\cdot)\) \(\chi_{9408}(4055,\cdot)\) \(\chi_{9408}(4295,\cdot)\) \(\chi_{9408}(4631,\cdot)\) \(\chi_{9408}(4727,\cdot)\) \(\chi_{9408}(5063,\cdot)\) \(\chi_{9408}(5303,\cdot)\) \(\chi_{9408}(5399,\cdot)\) \(\chi_{9408}(5639,\cdot)\) \(\chi_{9408}(5735,\cdot)\) ...

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: $\Q(\zeta_{168})$
Fixed field: Number field defined by a degree 168 polynomial (not computed)

Values on generators

\((1471,6469,3137,4609)\) → \((-1,e\left(\frac{5}{8}\right),-1,e\left(\frac{10}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 9408 }(5063, a) \) \(1\)\(1\)\(e\left(\frac{157}{168}\right)\)\(e\left(\frac{29}{168}\right)\)\(e\left(\frac{5}{56}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{71}{84}\right)\)\(e\left(\frac{73}{84}\right)\)\(e\left(\frac{53}{56}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{145}{168}\right)\)
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)
 
\( \chi_{ 9408 }(5063,a) \;\) at \(\;a = \) e.g. 2