Properties

Label 2268.13
Modulus $2268$
Conductor $567$
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([0,8,27]))
 
pari: [g,chi] = znchar(Mod(13,2268))
 

Basic properties

Modulus: \(2268\)
Conductor: \(567\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{567}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2268.cw

\(\chi_{2268}(13,\cdot)\) \(\chi_{2268}(97,\cdot)\) \(\chi_{2268}(265,\cdot)\) \(\chi_{2268}(349,\cdot)\) \(\chi_{2268}(517,\cdot)\) \(\chi_{2268}(601,\cdot)\) \(\chi_{2268}(769,\cdot)\) \(\chi_{2268}(853,\cdot)\) \(\chi_{2268}(1021,\cdot)\) \(\chi_{2268}(1105,\cdot)\) \(\chi_{2268}(1273,\cdot)\) \(\chi_{2268}(1357,\cdot)\) \(\chi_{2268}(1525,\cdot)\) \(\chi_{2268}(1609,\cdot)\) \(\chi_{2268}(1777,\cdot)\) \(\chi_{2268}(1861,\cdot)\) \(\chi_{2268}(2029,\cdot)\) \(\chi_{2268}(2113,\cdot)\)

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

\((1135,1541,325)\) → \((1,e\left(\frac{4}{27}\right),-1)\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{49}{54}\right)\)\(e\left(\frac{25}{27}\right)\)\(e\left(\frac{37}{54}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{17}{27}\right)\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{13}{27}\right)\)\(e\left(\frac{25}{54}\right)\)\(e\left(\frac{2}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2268 }(13,a) \;\) at \(\;a = \) e.g. 2