Properties

Label 1150.9
Modulus $1150$
Conductor $575$
Order $110$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, base_ring=CyclotomicField(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([77,50]))
 
pari: [g,chi] = znchar(Mod(9,1150))
 

Basic properties

Modulus: \(1150\)
Conductor: \(575\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{575}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1150.u

\(\chi_{1150}(9,\cdot)\) \(\chi_{1150}(29,\cdot)\) \(\chi_{1150}(39,\cdot)\) \(\chi_{1150}(59,\cdot)\) \(\chi_{1150}(119,\cdot)\) \(\chi_{1150}(169,\cdot)\) \(\chi_{1150}(179,\cdot)\) \(\chi_{1150}(209,\cdot)\) \(\chi_{1150}(219,\cdot)\) \(\chi_{1150}(239,\cdot)\) \(\chi_{1150}(259,\cdot)\) \(\chi_{1150}(269,\cdot)\) \(\chi_{1150}(279,\cdot)\) \(\chi_{1150}(289,\cdot)\) \(\chi_{1150}(409,\cdot)\) \(\chi_{1150}(439,\cdot)\) \(\chi_{1150}(469,\cdot)\) \(\chi_{1150}(489,\cdot)\) \(\chi_{1150}(509,\cdot)\) \(\chi_{1150}(519,\cdot)\) \(\chi_{1150}(579,\cdot)\) \(\chi_{1150}(629,\cdot)\) \(\chi_{1150}(639,\cdot)\) \(\chi_{1150}(669,\cdot)\) \(\chi_{1150}(679,\cdot)\) \(\chi_{1150}(719,\cdot)\) \(\chi_{1150}(729,\cdot)\) \(\chi_{1150}(739,\cdot)\) \(\chi_{1150}(809,\cdot)\) \(\chi_{1150}(859,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((277,51)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{5}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 1150 }(9, a) \) \(1\)\(1\)\(e\left(\frac{19}{110}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{19}{55}\right)\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{73}{110}\right)\)\(e\left(\frac{31}{110}\right)\)\(e\left(\frac{23}{55}\right)\)\(e\left(\frac{17}{55}\right)\)\(e\left(\frac{57}{110}\right)\)\(e\left(\frac{32}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1150 }(9,a) \;\) at \(\;a = \) e.g. 2