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Properties

Label 2201.402
Modulus $2201$
Conductor $2201$
Order $70$
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(2201, base_ring=CyclotomicField(70)) M = H._module chi = DirichletCharacter(H, M([35,9]))
 
Copy content gp:[g,chi] = znchar(Mod(402, 2201))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2201.402");
 

Basic properties

Modulus: \(2201\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2201\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(70\)
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 2201.cy

\(\chi_{2201}(61,\cdot)\) \(\chi_{2201}(92,\cdot)\) \(\chi_{2201}(123,\cdot)\) \(\chi_{2201}(278,\cdot)\) \(\chi_{2201}(340,\cdot)\) \(\chi_{2201}(402,\cdot)\) \(\chi_{2201}(433,\cdot)\) \(\chi_{2201}(495,\cdot)\) \(\chi_{2201}(650,\cdot)\) \(\chi_{2201}(681,\cdot)\) \(\chi_{2201}(743,\cdot)\) \(\chi_{2201}(836,\cdot)\) \(\chi_{2201}(991,\cdot)\) \(\chi_{2201}(1022,\cdot)\) \(\chi_{2201}(1053,\cdot)\) \(\chi_{2201}(1270,\cdot)\) \(\chi_{2201}(1487,\cdot)\) \(\chi_{2201}(1735,\cdot)\) \(\chi_{2201}(1766,\cdot)\) \(\chi_{2201}(1797,\cdot)\) \(\chi_{2201}(1828,\cdot)\) \(\chi_{2201}(1859,\cdot)\) \(\chi_{2201}(1890,\cdot)\) \(\chi_{2201}(1952,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((995,1427)\) → \((-1,e\left(\frac{9}{70}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2201 }(402, a) \) \(1\)\(1\)\(e\left(\frac{27}{35}\right)\)\(e\left(\frac{59}{70}\right)\)\(e\left(\frac{19}{35}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{43}{70}\right)\)\(e\left(\frac{9}{70}\right)\)\(e\left(\frac{11}{35}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{17}{35}\right)\)
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)
 
\( \chi_{ 2201 }(402,a) \;\) at \(\;a = \) e.g. 2