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Properties

Label 5625.1024
Modulus $5625$
Conductor $1125$
Order $150$
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(5625, base_ring=CyclotomicField(150)) M = H._module chi = DirichletCharacter(H, M([100,3]))
 
Copy content gp:[g,chi] = znchar(Mod(1024, 5625))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5625.1024");
 

Basic properties

Modulus: \(5625\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1125\)
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 \(\chi_{1125}(754,\cdot)\)
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 5625.bg

\(\chi_{5625}(49,\cdot)\) \(\chi_{5625}(274,\cdot)\) \(\chi_{5625}(349,\cdot)\) \(\chi_{5625}(574,\cdot)\) \(\chi_{5625}(724,\cdot)\) \(\chi_{5625}(799,\cdot)\) \(\chi_{5625}(949,\cdot)\) \(\chi_{5625}(1024,\cdot)\) \(\chi_{5625}(1174,\cdot)\) \(\chi_{5625}(1399,\cdot)\) \(\chi_{5625}(1474,\cdot)\) \(\chi_{5625}(1699,\cdot)\) \(\chi_{5625}(1849,\cdot)\) \(\chi_{5625}(1924,\cdot)\) \(\chi_{5625}(2074,\cdot)\) \(\chi_{5625}(2149,\cdot)\) \(\chi_{5625}(2299,\cdot)\) \(\chi_{5625}(2524,\cdot)\) \(\chi_{5625}(2599,\cdot)\) \(\chi_{5625}(2824,\cdot)\) \(\chi_{5625}(2974,\cdot)\) \(\chi_{5625}(3049,\cdot)\) \(\chi_{5625}(3199,\cdot)\) \(\chi_{5625}(3274,\cdot)\) \(\chi_{5625}(3424,\cdot)\) \(\chi_{5625}(3649,\cdot)\) \(\chi_{5625}(3724,\cdot)\) \(\chi_{5625}(3949,\cdot)\) \(\chi_{5625}(4099,\cdot)\) \(\chi_{5625}(4174,\cdot)\) ...

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

\((4376,1252)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{50}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 5625 }(1024, a) \) \(1\)\(1\)\(e\left(\frac{103}{150}\right)\)\(e\left(\frac{28}{75}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{14}{75}\right)\)\(e\left(\frac{17}{150}\right)\)\(e\left(\frac{4}{75}\right)\)\(e\left(\frac{56}{75}\right)\)\(e\left(\frac{23}{50}\right)\)\(e\left(\frac{9}{25}\right)\)
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)
 
\( \chi_{ 5625 }(1024,a) \;\) at \(\;a = \) e.g. 2