↑

Properties

Label 26136.6191
Modulus $26136$
Conductor $4356$
Order $330$
Real no
Primitive no
Minimal no
Parity even

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(26136, base_ring=CyclotomicField(330)) M = H._module chi = DirichletCharacter(H, M([165,0,55,228]))
 
Copy content gp:[g,chi] = znchar(Mod(6191, 26136))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("26136.6191");
 

Basic properties

Modulus: \(26136\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4356\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(330\)
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 \(\chi_{4356}(3287,\cdot)\)
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: no
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 26136.gq

\(\chi_{26136}(71,\cdot)\) \(\chi_{26136}(575,\cdot)\) \(\chi_{26136}(719,\cdot)\) \(\chi_{26136}(1367,\cdot)\) \(\chi_{26136}(1439,\cdot)\) \(\chi_{26136}(1655,\cdot)\) \(\chi_{26136}(2231,\cdot)\) \(\chi_{26136}(2303,\cdot)\) \(\chi_{26136}(2951,\cdot)\) \(\chi_{26136}(3095,\cdot)\) \(\chi_{26136}(3743,\cdot)\) \(\chi_{26136}(3815,\cdot)\) \(\chi_{26136}(4031,\cdot)\) \(\chi_{26136}(4823,\cdot)\) \(\chi_{26136}(5471,\cdot)\) \(\chi_{26136}(6119,\cdot)\) \(\chi_{26136}(6191,\cdot)\) \(\chi_{26136}(6407,\cdot)\) \(\chi_{26136}(6983,\cdot)\) \(\chi_{26136}(7055,\cdot)\) \(\chi_{26136}(7199,\cdot)\) \(\chi_{26136}(7703,\cdot)\) \(\chi_{26136}(7847,\cdot)\) \(\chi_{26136}(8495,\cdot)\) \(\chi_{26136}(8567,\cdot)\) \(\chi_{26136}(8783,\cdot)\) \(\chi_{26136}(9359,\cdot)\) \(\chi_{26136}(9431,\cdot)\) \(\chi_{26136}(9575,\cdot)\) \(\chi_{26136}(10079,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((6535,13069,19361,23113)\) → \((-1,1,e\left(\frac{1}{6}\right),e\left(\frac{38}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 26136 }(6191, a) \) \(1\)\(1\)\(e\left(\frac{317}{330}\right)\)\(e\left(\frac{1}{330}\right)\)\(e\left(\frac{19}{165}\right)\)\(e\left(\frac{39}{110}\right)\)\(e\left(\frac{93}{110}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{152}{165}\right)\)\(e\left(\frac{301}{330}\right)\)\(e\left(\frac{83}{330}\right)\)\(e\left(\frac{53}{55}\right)\)
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)
 
\( \chi_{ 26136 }(6191,a) \;\) at \(\;a = \) e.g. 2