Properties

Label 1089.40
Modulus $1089$
Conductor $99$
Order $30$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1089, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([10,21]))
 
pari: [g,chi] = znchar(Mod(40,1089))
 

Basic properties

Modulus: \(1089\)
Conductor: \(99\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{99}(40,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1089.s

\(\chi_{1089}(40,\cdot)\) \(\chi_{1089}(94,\cdot)\) \(\chi_{1089}(112,\cdot)\) \(\chi_{1089}(403,\cdot)\) \(\chi_{1089}(457,\cdot)\) \(\chi_{1089}(475,\cdot)\) \(\chi_{1089}(481,\cdot)\) \(\chi_{1089}(844,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.0.159386923550435671074967363509984324121230045171.1

Values on generators

\((848,244)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{10}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\(-1\)\(1\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{1}{10}\right)\)\(-1\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{3}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1089 }(40,a) \;\) at \(\;a = \) e.g. 2