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Properties

Label 13400.1173
Modulus $13400$
Conductor $13400$
Order $660$
Real no
Primitive yes
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(13400, base_ring=CyclotomicField(660)) M = H._module chi = DirichletCharacter(H, M([0,330,363,650]))
 
Copy content gp:[g,chi] = znchar(Mod(1173, 13400))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("13400.1173");
 

Basic properties

Modulus: \(13400\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(13400\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(660\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 13400.hd

\(\chi_{13400}(13,\cdot)\) \(\chi_{13400}(117,\cdot)\) \(\chi_{13400}(197,\cdot)\) \(\chi_{13400}(213,\cdot)\) \(\chi_{13400}(413,\cdot)\) \(\chi_{13400}(453,\cdot)\) \(\chi_{13400}(517,\cdot)\) \(\chi_{13400}(597,\cdot)\) \(\chi_{13400}(637,\cdot)\) \(\chi_{13400}(653,\cdot)\) \(\chi_{13400}(677,\cdot)\) \(\chi_{13400}(733,\cdot)\) \(\chi_{13400}(917,\cdot)\) \(\chi_{13400}(1037,\cdot)\) \(\chi_{13400}(1053,\cdot)\) \(\chi_{13400}(1133,\cdot)\) \(\chi_{13400}(1173,\cdot)\) \(\chi_{13400}(1213,\cdot)\) \(\chi_{13400}(1237,\cdot)\) \(\chi_{13400}(1317,\cdot)\) \(\chi_{13400}(1397,\cdot)\) \(\chi_{13400}(1453,\cdot)\)

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: $\Q(\zeta_{660})$
Fixed field: Number field defined by a degree 660 polynomial (not computed)

Values on generators

\((3351,6701,8577,13201)\) → \((1,-1,e\left(\frac{11}{20}\right),e\left(\frac{65}{66}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 13400 }(1173, a) \) \(1\)\(1\)\(e\left(\frac{167}{220}\right)\)\(e\left(\frac{53}{132}\right)\)\(e\left(\frac{57}{110}\right)\)\(e\left(\frac{67}{165}\right)\)\(e\left(\frac{437}{660}\right)\)\(e\left(\frac{119}{660}\right)\)\(e\left(\frac{41}{165}\right)\)\(e\left(\frac{53}{330}\right)\)\(e\left(\frac{413}{660}\right)\)\(e\left(\frac{61}{220}\right)\)
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)
 
\( \chi_{ 13400 }(1173,a) \;\) at \(\;a = \) e.g. 2