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Properties

Label 100129.31
Modulus $100129$
Conductor $100129$
Order $33376$
Real no
Primitive yes
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(100129, base_ring=CyclotomicField(33376)) M = H._module chi = DirichletCharacter(H, M([13099]))
 
Copy content gp:[g,chi] = znchar(Mod(31, 100129))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("100129.31");
 

Basic properties

Modulus: \(100129\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(100129\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(33376\)
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: yes
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 100129.bt

\(\chi_{100129}(31,\cdot)\) \(\chi_{100129}(33,\cdot)\) \(\chi_{100129}(47,\cdot)\) \(\chi_{100129}(57,\cdot)\) \(\chi_{100129}(62,\cdot)\) \(\chi_{100129}(66,\cdot)\) \(\chi_{100129}(94,\cdot)\) \(\chi_{100129}(114,\cdot)\) \(\chi_{100129}(115,\cdot)\) \(\chi_{100129}(132,\cdot)\) \(\chi_{100129}(143,\cdot)\) \(\chi_{100129}(145,\cdot)\) \(\chi_{100129}(159,\cdot)\) \(\chi_{100129}(193,\cdot)\) \(\chi_{100129}(205,\cdot)\) \(\chi_{100129}(207,\cdot)\) \(\chi_{100129}(217,\cdot)\) \(\chi_{100129}(227,\cdot)\) \(\chi_{100129}(230,\cdot)\) \(\chi_{100129}(231,\cdot)\) \(\chi_{100129}(247,\cdot)\) \(\chi_{100129}(248,\cdot)\) \(\chi_{100129}(261,\cdot)\) \(\chi_{100129}(264,\cdot)\) \(\chi_{100129}(271,\cdot)\) \(\chi_{100129}(275,\cdot)\) \(\chi_{100129}(281,\cdot)\) \(\chi_{100129}(286,\cdot)\) \(\chi_{100129}(290,\cdot)\) \(\chi_{100129}(303,\cdot)\) ...

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: $\Q(\zeta_{33376})$
Fixed field: Number field defined by a degree 33376 polynomial (not computed)

Values on generators

\(11\) → \(e\left(\frac{13099}{33376}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 100129 }(31, a) \) \(-1\)\(1\)\(e\left(\frac{856}{1043}\right)\)\(e\left(\frac{1817}{2384}\right)\)\(e\left(\frac{669}{1043}\right)\)\(e\left(\frac{7465}{8344}\right)\)\(e\left(\frac{9727}{16688}\right)\)\(e\left(\frac{1137}{1192}\right)\)\(e\left(\frac{482}{1043}\right)\)\(e\left(\frac{625}{1192}\right)\)\(e\left(\frac{5969}{8344}\right)\)\(e\left(\frac{13099}{33376}\right)\)
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)
 
\( \chi_{ 100129 }(31,a) \;\) at \(\;a = \) e.g. 2