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Properties

Label 38280.18517
Modulus $38280$
Conductor $12760$
Order $140$
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(38280, base_ring=CyclotomicField(140)) M = H._module chi = DirichletCharacter(H, M([0,70,0,35,28,135]))
 
Copy content gp:[g,chi] = znchar(Mod(18517, 38280))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("38280.18517");
 

Basic properties

Modulus: \(38280\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(12760\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(140\)
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 \(\chi_{12760}(5757,\cdot)\)
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 38280.xo

\(\chi_{38280}(37,\cdot)\) \(\chi_{38280}(1477,\cdot)\) \(\chi_{38280}(2077,\cdot)\) \(\chi_{38280}(2293,\cdot)\) \(\chi_{38280}(3733,\cdot)\) \(\chi_{38280}(4933,\cdot)\) \(\chi_{38280}(5173,\cdot)\) \(\chi_{38280}(5317,\cdot)\) \(\chi_{38280}(5773,\cdot)\) \(\chi_{38280}(8077,\cdot)\) \(\chi_{38280}(8413,\cdot)\) \(\chi_{38280}(8893,\cdot)\) \(\chi_{38280}(9277,\cdot)\) \(\chi_{38280}(10477,\cdot)\) \(\chi_{38280}(11917,\cdot)\) \(\chi_{38280}(12373,\cdot)\) \(\chi_{38280}(13093,\cdot)\) \(\chi_{38280}(13957,\cdot)\) \(\chi_{38280}(14173,\cdot)\) \(\chi_{38280}(15613,\cdot)\) \(\chi_{38280}(15757,\cdot)\) \(\chi_{38280}(15997,\cdot)\) \(\chi_{38280}(17653,\cdot)\) \(\chi_{38280}(18517,\cdot)\) \(\chi_{38280}(19237,\cdot)\) \(\chi_{38280}(19693,\cdot)\) \(\chi_{38280}(19717,\cdot)\) \(\chi_{38280}(22333,\cdot)\) \(\chi_{38280}(22357,\cdot)\) \(\chi_{38280}(23197,\cdot)\) ...

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: $\Q(\zeta_{140})$
Fixed field: Number field defined by a degree 140 polynomial (not computed)

Values on generators

\((28711,19141,12761,7657,10441,2641)\) → \((1,-1,1,i,e\left(\frac{1}{5}\right),e\left(\frac{27}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 38280 }(18517, a) \) \(1\)\(1\)\(e\left(\frac{31}{140}\right)\)\(e\left(\frac{113}{140}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{39}{140}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{23}{140}\right)\)\(e\left(\frac{3}{70}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{16}{35}\right)\)
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)
 
\( \chi_{ 38280 }(18517,a) \;\) at \(\;a = \) e.g. 2