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Properties

Label 8107.4480
Modulus $8107$
Conductor $737$
Order $110$
Real no
Primitive no
Minimal no
Parity odd

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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(8107, base_ring=CyclotomicField(110)) M = H._module chi = DirichletCharacter(H, M([88,75]))
 
Copy content gp:[g,chi] = znchar(Mod(4480, 8107))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8107.4480");
 

Basic properties

Modulus: \(8107\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(737\)
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 \(\chi_{737}(58,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 8107.ez

\(\chi_{8107}(3,\cdot)\) \(\chi_{8107}(27,\cdot)\) \(\chi_{8107}(444,\cdot)\) \(\chi_{8107}(511,\cdot)\) \(\chi_{8107}(608,\cdot)\) \(\chi_{8107}(807,\cdot)\) \(\chi_{8107}(856,\cdot)\) \(\chi_{8107}(874,\cdot)\) \(\chi_{8107}(1412,\cdot)\) \(\chi_{8107}(1479,\cdot)\) \(\chi_{8107}(2187,\cdot)\) \(\chi_{8107}(2665,\cdot)\) \(\chi_{8107}(2671,\cdot)\) \(\chi_{8107}(2792,\cdot)\) \(\chi_{8107}(3469,\cdot)\) \(\chi_{8107}(3536,\cdot)\) \(\chi_{8107}(3760,\cdot)\) \(\chi_{8107}(3996,\cdot)\) \(\chi_{8107}(4480,\cdot)\) \(\chi_{8107}(4601,\cdot)\) \(\chi_{8107}(4800,\cdot)\) \(\chi_{8107}(4867,\cdot)\) \(\chi_{8107}(5212,\cdot)\) \(\chi_{8107}(5284,\cdot)\) \(\chi_{8107}(5351,\cdot)\) \(\chi_{8107}(5405,\cdot)\) \(\chi_{8107}(5454,\cdot)\) \(\chi_{8107}(5472,\cdot)\) \(\chi_{8107}(5569,\cdot)\) \(\chi_{8107}(5938,\cdot)\) ...

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

\((3753,4357)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{15}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8107 }(4480, a) \) \(-1\)\(1\)\(e\left(\frac{53}{110}\right)\)\(e\left(\frac{109}{110}\right)\)\(e\left(\frac{53}{55}\right)\)\(e\left(\frac{47}{110}\right)\)\(e\left(\frac{26}{55}\right)\)\(e\left(\frac{31}{110}\right)\)\(e\left(\frac{49}{110}\right)\)\(e\left(\frac{54}{55}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{21}{22}\right)\)
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)
 
\( \chi_{ 8107 }(4480,a) \;\) at \(\;a = \) e.g. 2