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Properties

Label 39360.15637
Modulus $39360$
Conductor $13120$
Order $80$
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(39360, base_ring=CyclotomicField(80)) M = H._module chi = DirichletCharacter(H, M([0,65,0,20,48]))
 
Copy content gp:[g,chi] = znchar(Mod(15637, 39360))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("39360.15637");
 

Basic properties

Modulus: \(39360\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(13120\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(80\)
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 \(\chi_{13120}(2517,\cdot)\)
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 39360.bfu

\(\chi_{39360}(37,\cdot)\) \(\chi_{39360}(1453,\cdot)\) \(\chi_{39360}(2437,\cdot)\) \(\chi_{39360}(4813,\cdot)\) \(\chi_{39360}(5053,\cdot)\) \(\chi_{39360}(5797,\cdot)\) \(\chi_{39360}(6037,\cdot)\) \(\chi_{39360}(8893,\cdot)\) \(\chi_{39360}(9877,\cdot)\) \(\chi_{39360}(11293,\cdot)\) \(\chi_{39360}(12277,\cdot)\) \(\chi_{39360}(14653,\cdot)\) \(\chi_{39360}(14893,\cdot)\) \(\chi_{39360}(15637,\cdot)\) \(\chi_{39360}(15877,\cdot)\) \(\chi_{39360}(18733,\cdot)\) \(\chi_{39360}(19717,\cdot)\) \(\chi_{39360}(21133,\cdot)\) \(\chi_{39360}(22117,\cdot)\) \(\chi_{39360}(24493,\cdot)\) \(\chi_{39360}(24733,\cdot)\) \(\chi_{39360}(25477,\cdot)\) \(\chi_{39360}(25717,\cdot)\) \(\chi_{39360}(28573,\cdot)\) \(\chi_{39360}(29557,\cdot)\) \(\chi_{39360}(30973,\cdot)\) \(\chi_{39360}(31957,\cdot)\) \(\chi_{39360}(34333,\cdot)\) \(\chi_{39360}(34573,\cdot)\) \(\chi_{39360}(35317,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((11071,17221,13121,23617,21121)\) → \((1,e\left(\frac{13}{16}\right),1,i,e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(43\)
\( \chi_{ 39360 }(15637, a) \) \(-1\)\(1\)\(e\left(\frac{31}{40}\right)\)\(e\left(\frac{69}{80}\right)\)\(e\left(\frac{43}{80}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{47}{80}\right)\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{51}{80}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{61}{80}\right)\)\(e\left(\frac{73}{80}\right)\)
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)
 
\( \chi_{ 39360 }(15637,a) \;\) at \(\;a = \) e.g. 2