Properties

Label 2736.85
Modulus $2736$
Conductor $2736$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: 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([0,9,12,16]))
 
pari: [g,chi] = znchar(Mod(85,2736))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2736.gm

\(\chi_{2736}(85,\cdot)\) \(\chi_{2736}(517,\cdot)\) \(\chi_{2736}(853,\cdot)\) \(\chi_{2736}(1069,\cdot)\) \(\chi_{2736}(1165,\cdot)\) \(\chi_{2736}(1213,\cdot)\) \(\chi_{2736}(1453,\cdot)\) \(\chi_{2736}(1885,\cdot)\) \(\chi_{2736}(2221,\cdot)\) \(\chi_{2736}(2437,\cdot)\) \(\chi_{2736}(2533,\cdot)\) \(\chi_{2736}(2581,\cdot)\)

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: 36.36.4206346332526171592215690782651933064572033330381358894197631033983747859866315523440978690048.1

Values on generators

\((1711,2053,1217,1009)\) → \((1,i,e\left(\frac{1}{3}\right),e\left(\frac{4}{9}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\(1\)\(1\)\(e\left(\frac{1}{36}\right)\)\(-1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{19}{36}\right)\)
value at e.g. 2