Properties

Label 2268.29
Modulus $2268$
Conductor $81$
Order $54$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2268, base_ring=CyclotomicField(54))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,37,0]))
 
pari: [g,chi] = znchar(Mod(29,2268))
 

Basic properties

Modulus: \(2268\)
Conductor: \(81\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{81}(29,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2268.dc

\(\chi_{2268}(29,\cdot)\) \(\chi_{2268}(113,\cdot)\) \(\chi_{2268}(281,\cdot)\) \(\chi_{2268}(365,\cdot)\) \(\chi_{2268}(533,\cdot)\) \(\chi_{2268}(617,\cdot)\) \(\chi_{2268}(785,\cdot)\) \(\chi_{2268}(869,\cdot)\) \(\chi_{2268}(1037,\cdot)\) \(\chi_{2268}(1121,\cdot)\) \(\chi_{2268}(1289,\cdot)\) \(\chi_{2268}(1373,\cdot)\) \(\chi_{2268}(1541,\cdot)\) \(\chi_{2268}(1625,\cdot)\) \(\chi_{2268}(1793,\cdot)\) \(\chi_{2268}(1877,\cdot)\) \(\chi_{2268}(2045,\cdot)\) \(\chi_{2268}(2129,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 54 polynomial

Values on generators

\((1135,1541,325)\) → \((1,e\left(\frac{37}{54}\right),1)\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{41}{54}\right)\)\(e\left(\frac{49}{54}\right)\)\(e\left(\frac{13}{27}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{29}{54}\right)\)\(e\left(\frac{14}{27}\right)\)\(e\left(\frac{19}{54}\right)\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{7}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2268 }(29,a) \;\) at \(\;a = \) e.g. 2