↑

Properties

Label 6165.917
Modulus $6165$
Conductor $2055$
Order $136$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6165, base_ring=CyclotomicField(136)) M = H._module chi = DirichletCharacter(H, M([68,34,121]))
 
Copy content gp:[g,chi] = znchar(Mod(917, 6165))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6165.917");
 

Basic properties

Modulus: \(6165\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2055\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(136\)
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 \(\chi_{2055}(917,\cdot)\)
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 6165.dh

\(\chi_{6165}(53,\cdot)\) \(\chi_{6165}(143,\cdot)\) \(\chi_{6165}(188,\cdot)\) \(\chi_{6165}(287,\cdot)\) \(\chi_{6165}(332,\cdot)\) \(\chi_{6165}(458,\cdot)\) \(\chi_{6165}(602,\cdot)\) \(\chi_{6165}(638,\cdot)\) \(\chi_{6165}(737,\cdot)\) \(\chi_{6165}(782,\cdot)\) \(\chi_{6165}(908,\cdot)\) \(\chi_{6165}(917,\cdot)\) \(\chi_{6165}(953,\cdot)\) \(\chi_{6165}(962,\cdot)\) \(\chi_{6165}(1007,\cdot)\) \(\chi_{6165}(1043,\cdot)\) \(\chi_{6165}(1313,\cdot)\) \(\chi_{6165}(1322,\cdot)\) \(\chi_{6165}(1367,\cdot)\) \(\chi_{6165}(1403,\cdot)\) \(\chi_{6165}(1412,\cdot)\) \(\chi_{6165}(1547,\cdot)\) \(\chi_{6165}(1592,\cdot)\) \(\chi_{6165}(1673,\cdot)\) \(\chi_{6165}(1727,\cdot)\) \(\chi_{6165}(1808,\cdot)\) \(\chi_{6165}(1898,\cdot)\) \(\chi_{6165}(1988,\cdot)\) \(\chi_{6165}(1997,\cdot)\) \(\chi_{6165}(2042,\cdot)\) ...

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: $\Q(\zeta_{136})$
Fixed field: Number field defined by a degree 136 polynomial (not computed)

Values on generators

\((686,2467,6031)\) → \((-1,i,e\left(\frac{121}{136}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 6165 }(917, a) \) \(-1\)\(1\)\(e\left(\frac{11}{17}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{21}{34}\right)\)\(e\left(\frac{16}{17}\right)\)\(e\left(\frac{3}{68}\right)\)\(e\left(\frac{135}{136}\right)\)\(e\left(\frac{9}{34}\right)\)\(e\left(\frac{10}{17}\right)\)\(e\left(\frac{19}{34}\right)\)\(e\left(\frac{29}{68}\right)\)
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)
 
\( \chi_{ 6165 }(917,a) \;\) at \(\;a = \) e.g. 2