Properties

Label 9200.19
Modulus $9200$
Conductor $9200$
Order $220$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9200, base_ring=CyclotomicField(220))
 
M = H._module
 
chi = DirichletCharacter(H, M([110,165,198,150]))
 
pari: [g,chi] = znchar(Mod(19,9200))
 

Basic properties

Modulus: \(9200\)
Conductor: \(9200\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(220\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9200.fm

\(\chi_{9200}(19,\cdot)\) \(\chi_{9200}(339,\cdot)\) \(\chi_{9200}(379,\cdot)\) \(\chi_{9200}(419,\cdot)\) \(\chi_{9200}(539,\cdot)\) \(\chi_{9200}(619,\cdot)\) \(\chi_{9200}(659,\cdot)\) \(\chi_{9200}(779,\cdot)\) \(\chi_{9200}(819,\cdot)\) \(\chi_{9200}(939,\cdot)\) \(\chi_{9200}(1019,\cdot)\) \(\chi_{9200}(1259,\cdot)\) \(\chi_{9200}(1339,\cdot)\) \(\chi_{9200}(1459,\cdot)\) \(\chi_{9200}(1539,\cdot)\) \(\chi_{9200}(1579,\cdot)\) \(\chi_{9200}(1739,\cdot)\) \(\chi_{9200}(1859,\cdot)\) \(\chi_{9200}(1939,\cdot)\) \(\chi_{9200}(2179,\cdot)\) \(\chi_{9200}(2219,\cdot)\) \(\chi_{9200}(2259,\cdot)\) \(\chi_{9200}(2379,\cdot)\) \(\chi_{9200}(2459,\cdot)\) \(\chi_{9200}(2619,\cdot)\) \(\chi_{9200}(2659,\cdot)\) \(\chi_{9200}(2779,\cdot)\) \(\chi_{9200}(2859,\cdot)\) \(\chi_{9200}(3139,\cdot)\) \(\chi_{9200}(3179,\cdot)\) ...

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: $\Q(\zeta_{220})$
Fixed field: Number field defined by a degree 220 polynomial (not computed)

Values on generators

\((1151,6901,2577,1201)\) → \((-1,-i,e\left(\frac{9}{10}\right),e\left(\frac{15}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 9200 }(19, a) \) \(1\)\(1\)\(e\left(\frac{211}{220}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{101}{110}\right)\)\(e\left(\frac{173}{220}\right)\)\(e\left(\frac{197}{220}\right)\)\(e\left(\frac{26}{55}\right)\)\(e\left(\frac{39}{220}\right)\)\(e\left(\frac{91}{220}\right)\)\(e\left(\frac{193}{220}\right)\)\(e\left(\frac{71}{220}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9200 }(19,a) \;\) at \(\;a = \) e.g. 2