Properties

Label 9900.3097
Modulus $9900$
Conductor $275$
Order $20$
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(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,17,18]))
 
pari: [g,chi] = znchar(Mod(3097,9900))
 

Basic properties

Modulus: \(9900\)
Conductor: \(275\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{275}(72,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9900.ho

\(\chi_{9900}(73,\cdot)\) \(\chi_{9900}(613,\cdot)\) \(\chi_{9900}(1333,\cdot)\) \(\chi_{9900}(2917,\cdot)\) \(\chi_{9900}(3097,\cdot)\) \(\chi_{9900}(7453,\cdot)\) \(\chi_{9900}(8137,\cdot)\) \(\chi_{9900}(9577,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.20.16181489084533623771858401596546173095703125.1

Values on generators

\((4951,5501,2377,4501)\) → \((1,1,e\left(\frac{17}{20}\right),e\left(\frac{9}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 9900 }(3097, a) \) \(1\)\(1\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{3}{20}\right)\)\(1\)\(e\left(\frac{7}{20}\right)\)\(1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9900 }(3097,a) \;\) at \(\;a = \) e.g. 2