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Properties

Label 85547.49849
Modulus $85547$
Conductor $7777$
Order $150$
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(85547, base_ring=CyclotomicField(150)) M = H._module chi = DirichletCharacter(H, M([50,45,18]))
 
Copy content gp:[g,chi] = znchar(Mod(49849, 85547))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("85547.49849");
 

Basic properties

Modulus: \(85547\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(7777\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(150\)
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 \(\chi_{7777}(3187,\cdot)\)
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 85547.ms

\(\chi_{85547}(233,\cdot)\) \(\chi_{85547}(282,\cdot)\) \(\chi_{85547}(3742,\cdot)\) \(\chi_{85547}(3791,\cdot)\) \(\chi_{85547}(5926,\cdot)\) \(\chi_{85547}(7856,\cdot)\) \(\chi_{85547}(9774,\cdot)\) \(\chi_{85547}(17222,\cdot)\) \(\chi_{85547}(20325,\cdot)\) \(\chi_{85547}(21529,\cdot)\) \(\chi_{85547}(24917,\cdot)\) \(\chi_{85547}(25407,\cdot)\) \(\chi_{85547}(28184,\cdot)\) \(\chi_{85547}(28233,\cdot)\) \(\chi_{85547}(30368,\cdot)\) \(\chi_{85547}(36418,\cdot)\) \(\chi_{85547}(37265,\cdot)\) \(\chi_{85547}(40387,\cdot)\) \(\chi_{85547}(41906,\cdot)\) \(\chi_{85547}(44622,\cdot)\) \(\chi_{85547}(44767,\cdot)\) \(\chi_{85547}(45469,\cdot)\) \(\chi_{85547}(45971,\cdot)\) \(\chi_{85547}(49359,\cdot)\) \(\chi_{85547}(49849,\cdot)\) \(\chi_{85547}(57999,\cdot)\) \(\chi_{85547}(59021,\cdot)\) \(\chi_{85547}(60860,\cdot)\) \(\chi_{85547}(61338,\cdot)\) \(\chi_{85547}(61387,\cdot)\) ...

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: $\Q(\zeta_{75})$
Fixed field: Number field defined by a degree 150 polynomial (not computed)

Values on generators

\((61106,49491,48280)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{3}{10}\right),e\left(\frac{3}{25}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 85547 }(49849, a) \) \(-1\)\(1\)\(e\left(\frac{13}{150}\right)\)\(e\left(\frac{1}{75}\right)\)\(e\left(\frac{13}{75}\right)\)\(e\left(\frac{56}{75}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{13}{50}\right)\)\(e\left(\frac{2}{75}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{14}{75}\right)\)\(e\left(\frac{11}{50}\right)\)
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)
 
\( \chi_{ 85547 }(49849,a) \;\) at \(\;a = \) e.g. 2