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Properties

Label 14007.536
Modulus $14007$
Conductor $14007$
Order $924$
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(14007, base_ring=CyclotomicField(924)) M = H._module chi = DirichletCharacter(H, M([462,616,798,429]))
 
Copy content gp:[g,chi] = znchar(Mod(536, 14007))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("14007.536");
 

Basic properties

Modulus: \(14007\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(14007\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(924\)
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 14007.hg

\(\chi_{14007}(11,\cdot)\) \(\chi_{14007}(44,\cdot)\) \(\chi_{14007}(221,\cdot)\) \(\chi_{14007}(263,\cdot)\) \(\chi_{14007}(359,\cdot)\) \(\chi_{14007}(536,\cdot)\) \(\chi_{14007}(548,\cdot)\) \(\chi_{14007}(569,\cdot)\) \(\chi_{14007}(590,\cdot)\) \(\chi_{14007}(641,\cdot)\) \(\chi_{14007}(704,\cdot)\) \(\chi_{14007}(746,\cdot)\) \(\chi_{14007}(872,\cdot)\) \(\chi_{14007}(884,\cdot)\) \(\chi_{14007}(914,\cdot)\) \(\chi_{14007}(1052,\cdot)\) \(\chi_{14007}(1157,\cdot)\) \(\chi_{14007}(1178,\cdot)\) \(\chi_{14007}(1187,\cdot)\) \(\chi_{14007}(1229,\cdot)\) \(\chi_{14007}(1262,\cdot)\) \(\chi_{14007}(1355,\cdot)\) \(\chi_{14007}(1418,\cdot)\) \(\chi_{14007}(1460,\cdot)\) \(\chi_{14007}(1493,\cdot)\) \(\chi_{14007}(1523,\cdot)\) \(\chi_{14007}(1535,\cdot)\)

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: $\Q(\zeta_{924})$
Fixed field: Number field defined by a degree 924 polynomial (not computed)

Values on generators

\((4670,10006,9136,12559)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{19}{22}\right),e\left(\frac{13}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 14007 }(536, a) \) \(-1\)\(1\)\(e\left(\frac{23}{924}\right)\)\(e\left(\frac{23}{462}\right)\)\(e\left(\frac{421}{462}\right)\)\(e\left(\frac{23}{308}\right)\)\(e\left(\frac{865}{924}\right)\)\(e\left(\frac{505}{924}\right)\)\(e\left(\frac{69}{154}\right)\)\(e\left(\frac{23}{231}\right)\)\(e\left(\frac{127}{132}\right)\)\(e\left(\frac{431}{924}\right)\)
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)
 
\( \chi_{ 14007 }(536,a) \;\) at \(\;a = \) e.g. 2