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Properties

Label 8450.1739
Modulus $8450$
Conductor $4225$
Order $390$
Real no
Primitive no
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(8450, base_ring=CyclotomicField(390)) M = H._module chi = DirichletCharacter(H, M([117,145]))
 
Copy content gp:[g,chi] = znchar(Mod(1739, 8450))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8450.1739");
 

Basic properties

Modulus: \(8450\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4225\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(390\)
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 \(\chi_{4225}(1739,\cdot)\)
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: 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 8450.cv

\(\chi_{8450}(69,\cdot)\) \(\chi_{8450}(179,\cdot)\) \(\chi_{8450}(309,\cdot)\) \(\chi_{8450}(329,\cdot)\) \(\chi_{8450}(439,\cdot)\) \(\chi_{8450}(459,\cdot)\) \(\chi_{8450}(569,\cdot)\) \(\chi_{8450}(589,\cdot)\) \(\chi_{8450}(719,\cdot)\) \(\chi_{8450}(829,\cdot)\) \(\chi_{8450}(959,\cdot)\) \(\chi_{8450}(979,\cdot)\) \(\chi_{8450}(1089,\cdot)\) \(\chi_{8450}(1109,\cdot)\) \(\chi_{8450}(1219,\cdot)\) \(\chi_{8450}(1239,\cdot)\) \(\chi_{8450}(1369,\cdot)\) \(\chi_{8450}(1479,\cdot)\) \(\chi_{8450}(1609,\cdot)\) \(\chi_{8450}(1629,\cdot)\) \(\chi_{8450}(1739,\cdot)\) \(\chi_{8450}(1759,\cdot)\) \(\chi_{8450}(1869,\cdot)\) \(\chi_{8450}(1889,\cdot)\) \(\chi_{8450}(2019,\cdot)\) \(\chi_{8450}(2129,\cdot)\) \(\chi_{8450}(2259,\cdot)\) \(\chi_{8450}(2279,\cdot)\) \(\chi_{8450}(2409,\cdot)\) \(\chi_{8450}(2519,\cdot)\) ...

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: $\Q(\zeta_{195})$
Fixed field: Number field defined by a degree 390 polynomial (not computed)

Values on generators

\((677,3551)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{29}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 8450 }(1739, a) \) \(1\)\(1\)\(e\left(\frac{79}{390}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{79}{195}\right)\)\(e\left(\frac{37}{390}\right)\)\(e\left(\frac{71}{390}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{63}{130}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{79}{130}\right)\)\(e\left(\frac{92}{195}\right)\)
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)
 
\( \chi_{ 8450 }(1739,a) \;\) at \(\;a = \) e.g. 2