Properties

Label 9900.6673
Modulus $9900$
Conductor $2475$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9900, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,20,33,42]))
 
pari: [g,chi] = znchar(Mod(6673,9900))
 

Basic properties

Modulus: \(9900\)
Conductor: \(2475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2475}(1723,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9900.mv

\(\chi_{9900}(853,\cdot)\) \(\chi_{9900}(1537,\cdot)\) \(\chi_{9900}(2977,\cdot)\) \(\chi_{9900}(3373,\cdot)\) \(\chi_{9900}(3913,\cdot)\) \(\chi_{9900}(4153,\cdot)\) \(\chi_{9900}(4633,\cdot)\) \(\chi_{9900}(4837,\cdot)\) \(\chi_{9900}(6217,\cdot)\) \(\chi_{9900}(6277,\cdot)\) \(\chi_{9900}(6397,\cdot)\) \(\chi_{9900}(6673,\cdot)\) \(\chi_{9900}(7213,\cdot)\) \(\chi_{9900}(7933,\cdot)\) \(\chi_{9900}(9517,\cdot)\) \(\chi_{9900}(9697,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((4951,5501,2377,4501)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{11}{20}\right),e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9900 }(6673, a) \) \(1\)\(1\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{9}{20}\right)\)\(1\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{1}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9900 }(6673,a) \;\) at \(\;a = \) e.g. 2