↑

Properties

Label 4017.1790
Modulus $4017$
Conductor $4017$
Order $102$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4017, base_ring=CyclotomicField(102)) M = H._module chi = DirichletCharacter(H, M([51,68,9]))
 
Copy content gp:[g,chi] = znchar(Mod(1790, 4017))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4017.1790");
 

Basic properties

Modulus: \(4017\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4017\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(102\)
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 4017.dc

\(\chi_{4017}(113,\cdot)\) \(\chi_{4017}(230,\cdot)\) \(\chi_{4017}(542,\cdot)\) \(\chi_{4017}(698,\cdot)\) \(\chi_{4017}(731,\cdot)\) \(\chi_{4017}(815,\cdot)\) \(\chi_{4017}(848,\cdot)\) \(\chi_{4017}(893,\cdot)\) \(\chi_{4017}(1160,\cdot)\) \(\chi_{4017}(1316,\cdot)\) \(\chi_{4017}(1361,\cdot)\) \(\chi_{4017}(1433,\cdot)\) \(\chi_{4017}(1511,\cdot)\) \(\chi_{4017}(1634,\cdot)\) \(\chi_{4017}(1790,\cdot)\) \(\chi_{4017}(1979,\cdot)\) \(\chi_{4017}(2063,\cdot)\) \(\chi_{4017}(2102,\cdot)\) \(\chi_{4017}(2252,\cdot)\) \(\chi_{4017}(2258,\cdot)\) \(\chi_{4017}(2297,\cdot)\) \(\chi_{4017}(2408,\cdot)\) \(\chi_{4017}(2648,\cdot)\) \(\chi_{4017}(2681,\cdot)\) \(\chi_{4017}(2720,\cdot)\) \(\chi_{4017}(2876,\cdot)\) \(\chi_{4017}(2915,\cdot)\) \(\chi_{4017}(2921,\cdot)\) \(\chi_{4017}(3077,\cdot)\) \(\chi_{4017}(3266,\cdot)\) ...

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: $\Q(\zeta_{51})$
Fixed field: Number field defined by a degree 102 polynomial (not computed)

Values on generators

\((1340,1237,1756)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{3}{34}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\( \chi_{ 4017 }(1790, a) \) \(1\)\(1\)\(e\left(\frac{5}{102}\right)\)\(e\left(\frac{5}{51}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{35}{51}\right)\)\(e\left(\frac{5}{34}\right)\)\(e\left(\frac{65}{102}\right)\)\(e\left(\frac{28}{51}\right)\)\(e\left(\frac{25}{34}\right)\)\(e\left(\frac{10}{51}\right)\)\(e\left(\frac{1}{102}\right)\)
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)
 
\( \chi_{ 4017 }(1790,a) \;\) at \(\;a = \) e.g. 2