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Properties

Label 2900.1861
Modulus $2900$
Conductor $725$
Order $70$
Real no
Primitive no
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(2900, base_ring=CyclotomicField(70)) M = H._module chi = DirichletCharacter(H, M([0,56,55]))
 
Copy content gp:[g,chi] = znchar(Mod(1861, 2900))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2900.1861");
 

Basic properties

Modulus: \(2900\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(725\)
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: no, induced from \(\chi_{725}(411,\cdot)\)
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 2900.cj

\(\chi_{2900}(121,\cdot)\) \(\chi_{2900}(241,\cdot)\) \(\chi_{2900}(341,\cdot)\) \(\chi_{2900}(361,\cdot)\) \(\chi_{2900}(381,\cdot)\) \(\chi_{2900}(441,\cdot)\) \(\chi_{2900}(821,\cdot)\) \(\chi_{2900}(921,\cdot)\) \(\chi_{2900}(941,\cdot)\) \(\chi_{2900}(961,\cdot)\) \(\chi_{2900}(1021,\cdot)\) \(\chi_{2900}(1281,\cdot)\) \(\chi_{2900}(1521,\cdot)\) \(\chi_{2900}(1541,\cdot)\) \(\chi_{2900}(1861,\cdot)\) \(\chi_{2900}(1981,\cdot)\) \(\chi_{2900}(2081,\cdot)\) \(\chi_{2900}(2121,\cdot)\) \(\chi_{2900}(2181,\cdot)\) \(\chi_{2900}(2441,\cdot)\) \(\chi_{2900}(2561,\cdot)\) \(\chi_{2900}(2661,\cdot)\) \(\chi_{2900}(2681,\cdot)\) \(\chi_{2900}(2761,\cdot)\)

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

\((1451,1277,901)\) → \((1,e\left(\frac{4}{5}\right),e\left(\frac{11}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 2900 }(1861, a) \) \(1\)\(1\)\(e\left(\frac{37}{70}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{2}{35}\right)\)\(e\left(\frac{31}{70}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{33}{70}\right)\)\(e\left(\frac{67}{70}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{41}{70}\right)\)
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)
 
\( \chi_{ 2900 }(1861,a) \;\) at \(\;a = \) e.g. 2