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Properties

Label 5415.476
Modulus $5415$
Conductor $1083$
Order $38$
Real no
Primitive no
Minimal yes
Parity odd

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(5415, base_ring=CyclotomicField(38)) M = H._module chi = DirichletCharacter(H, M([19,0,4]))
 
Copy content gp:[g,chi] = znchar(Mod(476, 5415))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5415.476");
 

Basic properties

Modulus: \(5415\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1083\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(38\)
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 \(\chi_{1083}(476,\cdot)\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 5415.bp

\(\chi_{5415}(191,\cdot)\) \(\chi_{5415}(476,\cdot)\) \(\chi_{5415}(761,\cdot)\) \(\chi_{5415}(1046,\cdot)\) \(\chi_{5415}(1331,\cdot)\) \(\chi_{5415}(1616,\cdot)\) \(\chi_{5415}(1901,\cdot)\) \(\chi_{5415}(2186,\cdot)\) \(\chi_{5415}(2471,\cdot)\) \(\chi_{5415}(2756,\cdot)\) \(\chi_{5415}(3041,\cdot)\) \(\chi_{5415}(3326,\cdot)\) \(\chi_{5415}(3896,\cdot)\) \(\chi_{5415}(4181,\cdot)\) \(\chi_{5415}(4466,\cdot)\) \(\chi_{5415}(4751,\cdot)\) \(\chi_{5415}(5036,\cdot)\) \(\chi_{5415}(5321,\cdot)\)

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: \(\Q(\zeta_{19})\)
Fixed field: 38.0.136635360908492439649635218195335734184282612187547808802428015665989693822130747061222659537193769787.1

Values on generators

\((3611,2167,5056)\) → \((-1,1,e\left(\frac{2}{19}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(22\)
\( \chi_{ 5415 }(476, a) \) \(-1\)\(1\)\(e\left(\frac{23}{38}\right)\)\(e\left(\frac{4}{19}\right)\)\(e\left(\frac{15}{19}\right)\)\(e\left(\frac{31}{38}\right)\)\(e\left(\frac{9}{38}\right)\)\(e\left(\frac{12}{19}\right)\)\(e\left(\frac{15}{38}\right)\)\(e\left(\frac{8}{19}\right)\)\(e\left(\frac{11}{38}\right)\)\(e\left(\frac{16}{19}\right)\)
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)
 
\( \chi_{ 5415 }(476,a) \;\) at \(\;a = \) e.g. 2