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Properties

Label 24800.7203
Modulus $24800$
Conductor $24800$
Order $120$
Real no
Primitive yes
Minimal yes
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(24800, base_ring=CyclotomicField(120)) M = H._module chi = DirichletCharacter(H, M([60,45,42,92]))
 
Copy content gp:[g,chi] = znchar(Mod(7203, 24800))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("24800.7203");
 

Basic properties

Modulus: \(24800\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(24800\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(120\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 24800.bds

\(\chi_{24800}(827,\cdot)\) \(\chi_{24800}(2027,\cdot)\) \(\chi_{24800}(4083,\cdot)\) \(\chi_{24800}(4667,\cdot)\) \(\chi_{24800}(4963,\cdot)\) \(\chi_{24800}(7203,\cdot)\) \(\chi_{24800}(7523,\cdot)\) \(\chi_{24800}(7763,\cdot)\) \(\chi_{24800}(7947,\cdot)\) \(\chi_{24800}(8187,\cdot)\) \(\chi_{24800}(8723,\cdot)\) \(\chi_{24800}(8883,\cdot)\) \(\chi_{24800}(9603,\cdot)\) \(\chi_{24800}(10347,\cdot)\) \(\chi_{24800}(10987,\cdot)\) \(\chi_{24800}(12267,\cdot)\) \(\chi_{24800}(13227,\cdot)\) \(\chi_{24800}(14427,\cdot)\) \(\chi_{24800}(16483,\cdot)\) \(\chi_{24800}(17067,\cdot)\) \(\chi_{24800}(17363,\cdot)\) \(\chi_{24800}(19603,\cdot)\) \(\chi_{24800}(19923,\cdot)\) \(\chi_{24800}(20163,\cdot)\) \(\chi_{24800}(20347,\cdot)\) \(\chi_{24800}(20587,\cdot)\) \(\chi_{24800}(21123,\cdot)\) \(\chi_{24800}(21283,\cdot)\) \(\chi_{24800}(22003,\cdot)\) \(\chi_{24800}(22747,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((13951,21701,2977,8001)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{7}{20}\right),e\left(\frac{23}{30}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 24800 }(7203, a) \) \(-1\)\(1\)\(e\left(\frac{101}{120}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{73}{120}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{59}{120}\right)\)\(e\left(\frac{37}{120}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{21}{40}\right)\)
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)
 
\( \chi_{ 24800 }(7203,a) \;\) at \(\;a = \) e.g. 2