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Properties

Label 105125.31158
Modulus $105125$
Conductor $3625$
Order $100$
Real no
Primitive no
Minimal yes
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(105125, base_ring=CyclotomicField(100)) M = H._module chi = DirichletCharacter(H, M([83,25]))
 
Copy content gp:[g,chi] = znchar(Mod(31158, 105125))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("105125.31158");
 

Basic properties

Modulus: \(105125\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3625\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(100\)
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 \(\chi_{3625}(2158,\cdot)\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 105125.br

\(\chi_{105125}(1723,\cdot)\) \(\chi_{105125}(5928,\cdot)\) \(\chi_{105125}(6687,\cdot)\) \(\chi_{105125}(10133,\cdot)\) \(\chi_{105125}(10892,\cdot)\) \(\chi_{105125}(14338,\cdot)\) \(\chi_{105125}(15097,\cdot)\) \(\chi_{105125}(19302,\cdot)\) \(\chi_{105125}(22748,\cdot)\) \(\chi_{105125}(26953,\cdot)\) \(\chi_{105125}(27712,\cdot)\) \(\chi_{105125}(31158,\cdot)\) \(\chi_{105125}(31917,\cdot)\) \(\chi_{105125}(35363,\cdot)\) \(\chi_{105125}(36122,\cdot)\) \(\chi_{105125}(40327,\cdot)\) \(\chi_{105125}(43773,\cdot)\) \(\chi_{105125}(47978,\cdot)\) \(\chi_{105125}(48737,\cdot)\) \(\chi_{105125}(52183,\cdot)\) \(\chi_{105125}(52942,\cdot)\) \(\chi_{105125}(56388,\cdot)\) \(\chi_{105125}(57147,\cdot)\) \(\chi_{105125}(61352,\cdot)\) \(\chi_{105125}(64798,\cdot)\) \(\chi_{105125}(69003,\cdot)\) \(\chi_{105125}(69762,\cdot)\) \(\chi_{105125}(73208,\cdot)\) \(\chi_{105125}(73967,\cdot)\) \(\chi_{105125}(77413,\cdot)\) ...

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: $\Q(\zeta_{100})$
Fixed field: Number field defined by a degree 100 polynomial

Values on generators

\((9252,95876)\) → \((e\left(\frac{83}{100}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 105125 }(31158, a) \) \(1\)\(1\)\(e\left(\frac{2}{25}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{7}{50}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{3}{25}\right)\)\(e\left(\frac{33}{100}\right)\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{87}{100}\right)\)
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)
 
\( \chi_{ 105125 }(31158,a) \;\) at \(\;a = \) e.g. 2