Properties

Label 1089.977
Modulus $1089$
Conductor $99$
Order $30$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([25,18]))
 
pari: [g,chi] = znchar(Mod(977,1089))
 

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 \(\chi_{99}(86,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1089.r

\(\chi_{1089}(245,\cdot)\) \(\chi_{1089}(608,\cdot)\) \(\chi_{1089}(614,\cdot)\) \(\chi_{1089}(632,\cdot)\) \(\chi_{1089}(686,\cdot)\) \(\chi_{1089}(977,\cdot)\) \(\chi_{1089}(995,\cdot)\) \(\chi_{1089}(1049,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: 30.0.29099190400267368949073680941341991556317731763.1

Values on generators

\((848,244)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 1089 }(977, a) \) \(-1\)\(1\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{3}{10}\right)\)\(1\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{9}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1089 }(977,a) \;\) at \(\;a = \) e.g. 2