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Properties

Label 9200.6121
Modulus $9200$
Conductor $4600$
Order $110$
Real no
Primitive no
Minimal no
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(9200, base_ring=CyclotomicField(110)) M = H._module chi = DirichletCharacter(H, M([0,55,66,80]))
 
Copy content gp:[g,chi] = znchar(Mod(6121, 9200))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9200.6121");
 

Basic properties

Modulus: \(9200\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4600\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(110\)
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 \(\chi_{4600}(3821,\cdot)\)
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 9200.er

\(\chi_{9200}(41,\cdot)\) \(\chi_{9200}(121,\cdot)\) \(\chi_{9200}(361,\cdot)\) \(\chi_{9200}(441,\cdot)\) \(\chi_{9200}(761,\cdot)\) \(\chi_{9200}(841,\cdot)\) \(\chi_{9200}(1481,\cdot)\) \(\chi_{9200}(1641,\cdot)\) \(\chi_{9200}(1881,\cdot)\) \(\chi_{9200}(1961,\cdot)\) \(\chi_{9200}(2281,\cdot)\) \(\chi_{9200}(2441,\cdot)\) \(\chi_{9200}(2681,\cdot)\) \(\chi_{9200}(2841,\cdot)\) \(\chi_{9200}(3321,\cdot)\) \(\chi_{9200}(3481,\cdot)\) \(\chi_{9200}(3721,\cdot)\) \(\chi_{9200}(4041,\cdot)\) \(\chi_{9200}(4121,\cdot)\) \(\chi_{9200}(4281,\cdot)\) \(\chi_{9200}(4441,\cdot)\) \(\chi_{9200}(4521,\cdot)\) \(\chi_{9200}(4681,\cdot)\) \(\chi_{9200}(5161,\cdot)\) \(\chi_{9200}(5321,\cdot)\) \(\chi_{9200}(5561,\cdot)\) \(\chi_{9200}(5641,\cdot)\) \(\chi_{9200}(5881,\cdot)\) \(\chi_{9200}(5961,\cdot)\) \(\chi_{9200}(6121,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((1151,6901,2577,1201)\) → \((1,-1,e\left(\frac{3}{5}\right),e\left(\frac{8}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 9200 }(6121, a) \) \(1\)\(1\)\(e\left(\frac{37}{110}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{37}{55}\right)\)\(e\left(\frac{71}{110}\right)\)\(e\left(\frac{9}{110}\right)\)\(e\left(\frac{49}{55}\right)\)\(e\left(\frac{23}{110}\right)\)\(e\left(\frac{17}{110}\right)\)\(e\left(\frac{1}{110}\right)\)\(e\left(\frac{87}{110}\right)\)
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)
 
\( \chi_{ 9200 }(6121,a) \;\) at \(\;a = \) e.g. 2