Properties

Label 9200.471
Modulus $9200$
Conductor $4600$
Order $110$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9200, base_ring=CyclotomicField(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([55,55,66,45]))
 
pari: [g,chi] = znchar(Mod(471,9200))
 

Basic properties

Modulus: \(9200\)
Conductor: \(4600\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4600}(2771,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9200.ez

\(\chi_{9200}(471,\cdot)\) \(\chi_{9200}(631,\cdot)\) \(\chi_{9200}(711,\cdot)\) \(\chi_{9200}(871,\cdot)\) \(\chi_{9200}(1031,\cdot)\) \(\chi_{9200}(1111,\cdot)\) \(\chi_{9200}(1431,\cdot)\) \(\chi_{9200}(1671,\cdot)\) \(\chi_{9200}(1831,\cdot)\) \(\chi_{9200}(2311,\cdot)\) \(\chi_{9200}(2471,\cdot)\) \(\chi_{9200}(2711,\cdot)\) \(\chi_{9200}(2871,\cdot)\) \(\chi_{9200}(3191,\cdot)\) \(\chi_{9200}(3271,\cdot)\) \(\chi_{9200}(3511,\cdot)\) \(\chi_{9200}(3671,\cdot)\) \(\chi_{9200}(4311,\cdot)\) \(\chi_{9200}(4391,\cdot)\) \(\chi_{9200}(4711,\cdot)\) \(\chi_{9200}(4791,\cdot)\) \(\chi_{9200}(5031,\cdot)\) \(\chi_{9200}(5111,\cdot)\) \(\chi_{9200}(5511,\cdot)\) \(\chi_{9200}(5991,\cdot)\) \(\chi_{9200}(6231,\cdot)\) \(\chi_{9200}(6391,\cdot)\) \(\chi_{9200}(6631,\cdot)\) \(\chi_{9200}(6871,\cdot)\) \(\chi_{9200}(7191,\cdot)\) ...

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

\((1151,6901,2577,1201)\) → \((-1,-1,e\left(\frac{3}{5}\right),e\left(\frac{9}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 9200 }(471, a) \) \(1\)\(1\)\(e\left(\frac{41}{55}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{27}{55}\right)\)\(e\left(\frac{31}{110}\right)\)\(e\left(\frac{69}{110}\right)\)\(e\left(\frac{73}{110}\right)\)\(e\left(\frac{103}{110}\right)\)\(e\left(\frac{1}{55}\right)\)\(e\left(\frac{13}{55}\right)\)\(e\left(\frac{7}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9200 }(471,a) \;\) at \(\;a = \) e.g. 2