Properties

Label 2268.137
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([0,19,36]))
 
pari: [g,chi] = znchar(Mod(137,2268))
 

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 \(\chi_{567}(137,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2268.cp

\(\chi_{2268}(137,\cdot)\) \(\chi_{2268}(149,\cdot)\) \(\chi_{2268}(389,\cdot)\) \(\chi_{2268}(401,\cdot)\) \(\chi_{2268}(641,\cdot)\) \(\chi_{2268}(653,\cdot)\) \(\chi_{2268}(893,\cdot)\) \(\chi_{2268}(905,\cdot)\) \(\chi_{2268}(1145,\cdot)\) \(\chi_{2268}(1157,\cdot)\) \(\chi_{2268}(1397,\cdot)\) \(\chi_{2268}(1409,\cdot)\) \(\chi_{2268}(1649,\cdot)\) \(\chi_{2268}(1661,\cdot)\) \(\chi_{2268}(1901,\cdot)\) \(\chi_{2268}(1913,\cdot)\) \(\chi_{2268}(2153,\cdot)\) \(\chi_{2268}(2165,\cdot)\)

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

\((1135,1541,325)\) → \((1,e\left(\frac{19}{54}\right),e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{23}{54}\right)\)\(e\left(\frac{13}{54}\right)\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{11}{54}\right)\)\(e\left(\frac{23}{27}\right)\)\(e\left(\frac{1}{54}\right)\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{1}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2268 }(137,a) \;\) at \(\;a = \) e.g. 2