Properties

Label 9196.45
Modulus $9196$
Conductor $2299$
Order $33$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9196, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,18,22]))
 
pari: [g,chi] = znchar(Mod(45,9196))
 

Basic properties

Modulus: \(9196\)
Conductor: \(2299\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2299}(45,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9196.bw

\(\chi_{9196}(45,\cdot)\) \(\chi_{9196}(353,\cdot)\) \(\chi_{9196}(881,\cdot)\) \(\chi_{9196}(1189,\cdot)\) \(\chi_{9196}(1717,\cdot)\) \(\chi_{9196}(2025,\cdot)\) \(\chi_{9196}(2553,\cdot)\) \(\chi_{9196}(2861,\cdot)\) \(\chi_{9196}(3697,\cdot)\) \(\chi_{9196}(4225,\cdot)\) \(\chi_{9196}(4533,\cdot)\) \(\chi_{9196}(5061,\cdot)\) \(\chi_{9196}(5369,\cdot)\) \(\chi_{9196}(5897,\cdot)\) \(\chi_{9196}(6205,\cdot)\) \(\chi_{9196}(6733,\cdot)\) \(\chi_{9196}(7041,\cdot)\) \(\chi_{9196}(7569,\cdot)\) \(\chi_{9196}(7877,\cdot)\) \(\chi_{9196}(8405,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 33 polynomial

Values on generators

\((4599,3269,8229)\) → \((1,e\left(\frac{3}{11}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(21\)\(23\)\(25\)
\( \chi_{ 9196 }(45, a) \) \(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{1}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9196 }(45,a) \;\) at \(\;a = \) e.g. 2