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Properties

Label 4900.2689
Modulus $4900$
Conductor $1225$
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(4900, base_ring=CyclotomicField(70)) M = H._module chi = DirichletCharacter(H, M([0,21,10]))
 
Copy content gp:[g,chi] = znchar(Mod(2689, 4900))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4900.2689");
 

Basic properties

Modulus: \(4900\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1225\)
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 \(\chi_{1225}(239,\cdot)\)
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 4900.ct

\(\chi_{4900}(29,\cdot)\) \(\chi_{4900}(169,\cdot)\) \(\chi_{4900}(309,\cdot)\) \(\chi_{4900}(729,\cdot)\) \(\chi_{4900}(869,\cdot)\) \(\chi_{4900}(1009,\cdot)\) \(\chi_{4900}(1289,\cdot)\) \(\chi_{4900}(1429,\cdot)\) \(\chi_{4900}(1709,\cdot)\) \(\chi_{4900}(1989,\cdot)\) \(\chi_{4900}(2129,\cdot)\) \(\chi_{4900}(2269,\cdot)\) \(\chi_{4900}(2409,\cdot)\) \(\chi_{4900}(2689,\cdot)\) \(\chi_{4900}(2829,\cdot)\) \(\chi_{4900}(2969,\cdot)\) \(\chi_{4900}(3109,\cdot)\) \(\chi_{4900}(3389,\cdot)\) \(\chi_{4900}(3669,\cdot)\) \(\chi_{4900}(3809,\cdot)\) \(\chi_{4900}(4089,\cdot)\) \(\chi_{4900}(4229,\cdot)\) \(\chi_{4900}(4369,\cdot)\) \(\chi_{4900}(4789,\cdot)\)

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

\((2451,1177,101)\) → \((1,e\left(\frac{3}{10}\right),e\left(\frac{1}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 4900 }(2689, a) \) \(1\)\(1\)\(e\left(\frac{17}{70}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{29}{70}\right)\)\(e\left(\frac{33}{70}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{51}{70}\right)\)\(e\left(\frac{51}{70}\right)\)\(e\left(\frac{6}{35}\right)\)\(e\left(\frac{2}{5}\right)\)
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)
 
\( \chi_{ 4900 }(2689,a) \;\) at \(\;a = \) e.g. 2