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Properties

Label 1111.843
Modulus $1111$
Conductor $1111$
Order $100$
Real no
Primitive yes
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(1111, base_ring=CyclotomicField(100)) M = H._module chi = DirichletCharacter(H, M([70,33]))
 
Copy content gp:[g,chi] = znchar(Mod(843, 1111))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1111.843");
 

Basic properties

Modulus: \(1111\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1111\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1111.cn

\(\chi_{1111}(28,\cdot)\) \(\chi_{1111}(63,\cdot)\) \(\chi_{1111}(72,\cdot)\) \(\chi_{1111}(74,\cdot)\) \(\chi_{1111}(162,\cdot)\) \(\chi_{1111}(194,\cdot)\) \(\chi_{1111}(204,\cdot)\) \(\chi_{1111}(228,\cdot)\) \(\chi_{1111}(244,\cdot)\) \(\chi_{1111}(310,\cdot)\) \(\chi_{1111}(321,\cdot)\) \(\chi_{1111}(349,\cdot)\) \(\chi_{1111}(354,\cdot)\) \(\chi_{1111}(402,\cdot)\) \(\chi_{1111}(459,\cdot)\) \(\chi_{1111}(508,\cdot)\) \(\chi_{1111}(513,\cdot)\) \(\chi_{1111}(534,\cdot)\) \(\chi_{1111}(578,\cdot)\) \(\chi_{1111}(580,\cdot)\) \(\chi_{1111}(646,\cdot)\) \(\chi_{1111}(722,\cdot)\) \(\chi_{1111}(734,\cdot)\) \(\chi_{1111}(745,\cdot)\) \(\chi_{1111}(755,\cdot)\) \(\chi_{1111}(805,\cdot)\) \(\chi_{1111}(820,\cdot)\) \(\chi_{1111}(842,\cdot)\) \(\chi_{1111}(843,\cdot)\) \(\chi_{1111}(875,\cdot)\) ...

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

\((607,507)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{33}{100}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1111 }(843, a) \) \(1\)\(1\)\(e\left(\frac{3}{100}\right)\)\(e\left(\frac{37}{100}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{87}{100}\right)\)\(e\left(\frac{9}{100}\right)\)\(e\left(\frac{37}{50}\right)\)\(-i\)\(e\left(\frac{43}{100}\right)\)
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)
 
\( \chi_{ 1111 }(843,a) \;\) at \(\;a = \) e.g. 2