Properties

Label 9450.977
Modulus $9450$
Conductor $4725$
Order $180$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9450, base_ring=CyclotomicField(180))
 
M = H._module
 
chi = DirichletCharacter(H, M([50,9,120]))
 
pari: [g,chi] = znchar(Mod(977,9450))
 

Basic properties

Modulus: \(9450\)
Conductor: \(4725\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4725}(977,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9450.ha

\(\chi_{9450}(317,\cdot)\) \(\chi_{9450}(347,\cdot)\) \(\chi_{9450}(473,\cdot)\) \(\chi_{9450}(947,\cdot)\) \(\chi_{9450}(977,\cdot)\) \(\chi_{9450}(1073,\cdot)\) \(\chi_{9450}(1103,\cdot)\) \(\chi_{9450}(1577,\cdot)\) \(\chi_{9450}(1703,\cdot)\) \(\chi_{9450}(1733,\cdot)\) \(\chi_{9450}(2237,\cdot)\) \(\chi_{9450}(2333,\cdot)\) \(\chi_{9450}(2363,\cdot)\) \(\chi_{9450}(2837,\cdot)\) \(\chi_{9450}(2867,\cdot)\) \(\chi_{9450}(2963,\cdot)\) \(\chi_{9450}(3467,\cdot)\) \(\chi_{9450}(3497,\cdot)\) \(\chi_{9450}(3623,\cdot)\) \(\chi_{9450}(4097,\cdot)\) \(\chi_{9450}(4127,\cdot)\) \(\chi_{9450}(4223,\cdot)\) \(\chi_{9450}(4253,\cdot)\) \(\chi_{9450}(4727,\cdot)\) \(\chi_{9450}(4853,\cdot)\) \(\chi_{9450}(4883,\cdot)\) \(\chi_{9450}(5387,\cdot)\) \(\chi_{9450}(5483,\cdot)\) \(\chi_{9450}(5513,\cdot)\) \(\chi_{9450}(5987,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((9101,6427,6751)\) → \((e\left(\frac{5}{18}\right),e\left(\frac{1}{20}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9450 }(977, a) \) \(1\)\(1\)\(e\left(\frac{7}{90}\right)\)\(e\left(\frac{31}{180}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{169}{180}\right)\)\(e\left(\frac{17}{45}\right)\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{83}{90}\right)\)\(e\left(\frac{31}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9450 }(977,a) \;\) at \(\;a = \) e.g. 2