Properties

Label 2736.205
Modulus $2736$
Conductor $2736$
Order $36$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2736, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,27,24,22]))
 
pari: [g,chi] = znchar(Mod(205,2736))
 

Basic properties

Modulus: \(2736\)
Conductor: \(2736\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2736.gy

\(\chi_{2736}(205,\cdot)\) \(\chi_{2736}(637,\cdot)\) \(\chi_{2736}(877,\cdot)\) \(\chi_{2736}(925,\cdot)\) \(\chi_{2736}(1021,\cdot)\) \(\chi_{2736}(1237,\cdot)\) \(\chi_{2736}(1573,\cdot)\) \(\chi_{2736}(2005,\cdot)\) \(\chi_{2736}(2245,\cdot)\) \(\chi_{2736}(2293,\cdot)\) \(\chi_{2736}(2389,\cdot)\) \(\chi_{2736}(2605,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: 36.0.1518491026041947944789864372537347836310504032267670560805344803268132977411739903962193307107328.1

Values on generators

\((1711,2053,1217,1009)\) → \((1,-i,e\left(\frac{2}{3}\right),e\left(\frac{11}{18}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(-1\)\(1\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{5}{6}\right)\)\(-i\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{11}{36}\right)\)\(-1\)\(e\left(\frac{25}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2736 }(205,a) \;\) at \(\;a = \) e.g. 2