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Properties

Label 7803.5044
Modulus $7803$
Conductor $459$
Order $144$
Real no
Primitive no
Minimal yes
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(7803, base_ring=CyclotomicField(144)) M = H._module chi = DirichletCharacter(H, M([112,117]))
 
Copy content gp:[g,chi] = znchar(Mod(5044, 7803))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7803.5044");
 

Basic properties

Modulus: \(7803\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(459\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(144\)
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 \(\chi_{459}(454,\cdot)\)
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: 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 7803.bo

\(\chi_{7803}(40,\cdot)\) \(\chi_{7803}(214,\cdot)\) \(\chi_{7803}(364,\cdot)\) \(\chi_{7803}(538,\cdot)\) \(\chi_{7803}(643,\cdot)\) \(\chi_{7803}(709,\cdot)\) \(\chi_{7803}(736,\cdot)\) \(\chi_{7803}(907,\cdot)\) \(\chi_{7803}(1231,\cdot)\) \(\chi_{7803}(1510,\cdot)\) \(\chi_{7803}(1669,\cdot)\) \(\chi_{7803}(1948,\cdot)\) \(\chi_{7803}(2272,\cdot)\) \(\chi_{7803}(2443,\cdot)\) \(\chi_{7803}(2470,\cdot)\) \(\chi_{7803}(2536,\cdot)\) \(\chi_{7803}(2641,\cdot)\) \(\chi_{7803}(2815,\cdot)\) \(\chi_{7803}(2965,\cdot)\) \(\chi_{7803}(3139,\cdot)\) \(\chi_{7803}(3244,\cdot)\) \(\chi_{7803}(3310,\cdot)\) \(\chi_{7803}(3337,\cdot)\) \(\chi_{7803}(3508,\cdot)\) \(\chi_{7803}(3832,\cdot)\) \(\chi_{7803}(4111,\cdot)\) \(\chi_{7803}(4270,\cdot)\) \(\chi_{7803}(4549,\cdot)\) \(\chi_{7803}(4873,\cdot)\) \(\chi_{7803}(5044,\cdot)\) ...

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: $\Q(\zeta_{144})$
Fixed field: Number field defined by a degree 144 polynomial (not computed)

Values on generators

\((2891,2026)\) → \((e\left(\frac{7}{9}\right),e\left(\frac{13}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 7803 }(5044, a) \) \(-1\)\(1\)\(e\left(\frac{11}{72}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{137}{144}\right)\)\(e\left(\frac{55}{144}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{5}{48}\right)\)\(e\left(\frac{115}{144}\right)\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{77}{144}\right)\)\(e\left(\frac{11}{18}\right)\)
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)
 
\( \chi_{ 7803 }(5044,a) \;\) at \(\;a = \) e.g. 2