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Properties

Label 14720.1139
Modulus $14720$
Conductor $14720$
Order $352$
Real no
Primitive yes
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(14720, base_ring=CyclotomicField(352)) M = H._module chi = DirichletCharacter(H, M([176,165,176,320]))
 
Copy content gp:[g,chi] = znchar(Mod(1139, 14720))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("14720.1139");
 

Basic properties

Modulus: \(14720\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(14720\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(352\)
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: yes
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 14720.gm

\(\chi_{14720}(59,\cdot)\) \(\chi_{14720}(179,\cdot)\) \(\chi_{14720}(219,\cdot)\) \(\chi_{14720}(259,\cdot)\) \(\chi_{14720}(499,\cdot)\) \(\chi_{14720}(579,\cdot)\) \(\chi_{14720}(699,\cdot)\) \(\chi_{14720}(739,\cdot)\) \(\chi_{14720}(859,\cdot)\) \(\chi_{14720}(899,\cdot)\) \(\chi_{14720}(979,\cdot)\) \(\chi_{14720}(1099,\cdot)\) \(\chi_{14720}(1139,\cdot)\) \(\chi_{14720}(1179,\cdot)\) \(\chi_{14720}(1419,\cdot)\) \(\chi_{14720}(1499,\cdot)\) \(\chi_{14720}(1619,\cdot)\) \(\chi_{14720}(1659,\cdot)\) \(\chi_{14720}(1779,\cdot)\) \(\chi_{14720}(1819,\cdot)\) \(\chi_{14720}(1899,\cdot)\) \(\chi_{14720}(2019,\cdot)\) \(\chi_{14720}(2059,\cdot)\) \(\chi_{14720}(2099,\cdot)\) \(\chi_{14720}(2339,\cdot)\) \(\chi_{14720}(2419,\cdot)\) \(\chi_{14720}(2539,\cdot)\) \(\chi_{14720}(2579,\cdot)\) \(\chi_{14720}(2699,\cdot)\) \(\chi_{14720}(2739,\cdot)\) ...

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: $\Q(\zeta_{352})$
Fixed field: Number field defined by a degree 352 polynomial (not computed)

Values on generators

\((1151,12421,11777,14081)\) → \((-1,e\left(\frac{15}{32}\right),-1,e\left(\frac{10}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 14720 }(1139, a) \) \(-1\)\(1\)\(e\left(\frac{335}{352}\right)\)\(e\left(\frac{169}{176}\right)\)\(e\left(\frac{159}{176}\right)\)\(e\left(\frac{185}{352}\right)\)\(e\left(\frac{91}{352}\right)\)\(e\left(\frac{87}{88}\right)\)\(e\left(\frac{323}{352}\right)\)\(e\left(\frac{321}{352}\right)\)\(e\left(\frac{301}{352}\right)\)\(e\left(\frac{7}{352}\right)\)
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)
 
\( \chi_{ 14720 }(1139,a) \;\) at \(\;a = \) e.g. 2