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Properties

Label 100391.101
Modulus $100391$
Conductor $100391$
Order $100390$
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(100391, base_ring=CyclotomicField(100390)) M = H._module chi = DirichletCharacter(H, M([48789]))
 
Copy content gp:[g,chi] = znchar(Mod(101, 100391))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("100391.101");
 

Basic properties

Modulus: \(100391\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(100391\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(100390\)
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 100391.h

\(\chi_{100391}(7,\cdot)\) \(\chi_{100391}(11,\cdot)\) \(\chi_{100391}(13,\cdot)\) \(\chi_{100391}(14,\cdot)\) \(\chi_{100391}(21,\cdot)\) \(\chi_{100391}(22,\cdot)\) \(\chi_{100391}(26,\cdot)\) \(\chi_{100391}(33,\cdot)\) \(\chi_{100391}(34,\cdot)\) \(\chi_{100391}(35,\cdot)\) \(\chi_{100391}(39,\cdot)\) \(\chi_{100391}(42,\cdot)\) \(\chi_{100391}(44,\cdot)\) \(\chi_{100391}(51,\cdot)\) \(\chi_{100391}(55,\cdot)\) \(\chi_{100391}(56,\cdot)\) \(\chi_{100391}(63,\cdot)\) \(\chi_{100391}(65,\cdot)\) \(\chi_{100391}(66,\cdot)\) \(\chi_{100391}(68,\cdot)\) \(\chi_{100391}(78,\cdot)\) \(\chi_{100391}(83,\cdot)\) \(\chi_{100391}(84,\cdot)\) \(\chi_{100391}(85,\cdot)\) \(\chi_{100391}(101,\cdot)\) \(\chi_{100391}(104,\cdot)\) \(\chi_{100391}(105,\cdot)\) \(\chi_{100391}(107,\cdot)\) \(\chi_{100391}(109,\cdot)\) \(\chi_{100391}(110,\cdot)\) ...

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: $\Q(\zeta_{50195})$
Fixed field: Number field defined by a degree 100390 polynomial (not computed)

Values on generators

\(7\) → \(e\left(\frac{48789}{100390}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 100391 }(101, a) \) \(-1\)\(1\)\(e\left(\frac{47549}{50195}\right)\)\(e\left(\frac{33726}{50195}\right)\)\(e\left(\frac{44903}{50195}\right)\)\(e\left(\frac{26054}{50195}\right)\)\(e\left(\frac{6216}{10039}\right)\)\(e\left(\frac{48789}{100390}\right)\)\(e\left(\frac{42257}{50195}\right)\)\(e\left(\frac{17257}{50195}\right)\)\(e\left(\frac{23408}{50195}\right)\)\(e\left(\frac{61801}{100390}\right)\)
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)
 
\( \chi_{ 100391 }(101,a) \;\) at \(\;a = \) e.g. 2