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Properties

Label 2005.998
Modulus $2005$
Conductor $2005$
Order $100$
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(2005, base_ring=CyclotomicField(100)) M = H._module chi = DirichletCharacter(H, M([75,44]))
 
Copy content gp:[g,chi] = znchar(Mod(998, 2005))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2005.998");
 

Basic properties

Modulus: \(2005\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2005\)
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: 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 2005.bp

\(\chi_{2005}(63,\cdot)\) \(\chi_{2005}(77,\cdot)\) \(\chi_{2005}(88,\cdot)\) \(\chi_{2005}(173,\cdot)\) \(\chi_{2005}(178,\cdot)\) \(\chi_{2005}(387,\cdot)\) \(\chi_{2005}(452,\cdot)\) \(\chi_{2005}(478,\cdot)\) \(\chi_{2005}(597,\cdot)\) \(\chi_{2005}(657,\cdot)\) \(\chi_{2005}(722,\cdot)\) \(\chi_{2005}(732,\cdot)\) \(\chi_{2005}(788,\cdot)\) \(\chi_{2005}(807,\cdot)\) \(\chi_{2005}(827,\cdot)\) \(\chi_{2005}(853,\cdot)\) \(\chi_{2005}(927,\cdot)\) \(\chi_{2005}(997,\cdot)\) \(\chi_{2005}(998,\cdot)\) \(\chi_{2005}(1057,\cdot)\) \(\chi_{2005}(1058,\cdot)\) \(\chi_{2005}(1117,\cdot)\) \(\chi_{2005}(1123,\cdot)\) \(\chi_{2005}(1133,\cdot)\) \(\chi_{2005}(1162,\cdot)\) \(\chi_{2005}(1187,\cdot)\) \(\chi_{2005}(1208,\cdot)\) \(\chi_{2005}(1228,\cdot)\) \(\chi_{2005}(1328,\cdot)\) \(\chi_{2005}(1398,\cdot)\) ...

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

\((402,1206)\) → \((-i,e\left(\frac{11}{25}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 2005 }(998, a) \) \(-1\)\(1\)\(e\left(\frac{19}{100}\right)\)\(e\left(\frac{69}{100}\right)\)\(e\left(\frac{19}{50}\right)\)\(e\left(\frac{22}{25}\right)\)\(e\left(\frac{3}{100}\right)\)\(e\left(\frac{57}{100}\right)\)\(e\left(\frac{19}{50}\right)\)\(e\left(\frac{9}{25}\right)\)\(e\left(\frac{7}{100}\right)\)\(e\left(\frac{61}{100}\right)\)
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)
 
\( \chi_{ 2005 }(998,a) \;\) at \(\;a = \) e.g. 2