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Properties

Label 88305.6826
Modulus $88305$
Conductor $841$
Order $812$
Real no
Primitive no
Minimal yes
Parity odd

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(88305, base_ring=CyclotomicField(812)) M = H._module chi = DirichletCharacter(H, M([0,0,0,753]))
 
Copy content gp:[g,chi] = znchar(Mod(6826, 88305))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("88305.6826");
 

Basic properties

Modulus: \(88305\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(841\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(812\)
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 \(\chi_{841}(98,\cdot)\)
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 88305.kg

\(\chi_{88305}(106,\cdot)\) \(\chi_{88305}(211,\cdot)\) \(\chi_{88305}(316,\cdot)\) \(\chi_{88305}(421,\cdot)\) \(\chi_{88305}(736,\cdot)\) \(\chi_{88305}(946,\cdot)\) \(\chi_{88305}(1261,\cdot)\) \(\chi_{88305}(1366,\cdot)\) \(\chi_{88305}(1471,\cdot)\) \(\chi_{88305}(1576,\cdot)\) \(\chi_{88305}(2206,\cdot)\) \(\chi_{88305}(2521,\cdot)\) \(\chi_{88305}(3151,\cdot)\) \(\chi_{88305}(3256,\cdot)\) \(\chi_{88305}(3361,\cdot)\) \(\chi_{88305}(3466,\cdot)\) \(\chi_{88305}(3781,\cdot)\) \(\chi_{88305}(3991,\cdot)\) \(\chi_{88305}(4306,\cdot)\) \(\chi_{88305}(4411,\cdot)\) \(\chi_{88305}(4516,\cdot)\) \(\chi_{88305}(5251,\cdot)\) \(\chi_{88305}(5566,\cdot)\) \(\chi_{88305}(6196,\cdot)\) \(\chi_{88305}(6301,\cdot)\) \(\chi_{88305}(6406,\cdot)\) \(\chi_{88305}(6511,\cdot)\) \(\chi_{88305}(6826,\cdot)\) \(\chi_{88305}(7036,\cdot)\) \(\chi_{88305}(7351,\cdot)\) ...

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: $\Q(\zeta_{812})$
Fixed field: Number field defined by a degree 812 polynomial (not computed)

Values on generators

\((58871,17662,25231,87466)\) → \((1,1,1,e\left(\frac{753}{812}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 88305 }(6826, a) \) \(-1\)\(1\)\(e\left(\frac{753}{812}\right)\)\(e\left(\frac{347}{406}\right)\)\(e\left(\frac{635}{812}\right)\)\(e\left(\frac{93}{812}\right)\)\(e\left(\frac{365}{406}\right)\)\(e\left(\frac{144}{203}\right)\)\(e\left(\frac{107}{116}\right)\)\(e\left(\frac{477}{812}\right)\)\(e\left(\frac{17}{406}\right)\)\(e\left(\frac{118}{203}\right)\)
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)
 
\( \chi_{ 88305 }(6826,a) \;\) at \(\;a = \) e.g. 2