Properties

Label 9200.243
Modulus $9200$
Conductor $1840$
Order $44$
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(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,33,33,28]))
 
pari: [g,chi] = znchar(Mod(243,9200))
 

Basic properties

Modulus: \(9200\)
Conductor: \(1840\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1840}(243,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9200.dv

\(\chi_{9200}(243,\cdot)\) \(\chi_{9200}(1043,\cdot)\) \(\chi_{9200}(1843,\cdot)\) \(\chi_{9200}(2243,\cdot)\) \(\chi_{9200}(2907,\cdot)\) \(\chi_{9200}(3307,\cdot)\) \(\chi_{9200}(3443,\cdot)\) \(\chi_{9200}(3707,\cdot)\) \(\chi_{9200}(3843,\cdot)\) \(\chi_{9200}(4107,\cdot)\) \(\chi_{9200}(4907,\cdot)\) \(\chi_{9200}(5043,\cdot)\) \(\chi_{9200}(5707,\cdot)\) \(\chi_{9200}(6107,\cdot)\) \(\chi_{9200}(7307,\cdot)\) \(\chi_{9200}(7707,\cdot)\) \(\chi_{9200}(8243,\cdot)\) \(\chi_{9200}(8643,\cdot)\) \(\chi_{9200}(8907,\cdot)\) \(\chi_{9200}(9043,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((1151,6901,2577,1201)\) → \((-1,-i,-i,e\left(\frac{7}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 9200 }(243, a) \) \(1\)\(1\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{9}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9200 }(243,a) \;\) at \(\;a = \) e.g. 2