Properties

Label 3850.97
Modulus $3850$
Conductor $1925$
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(3850, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([17,10,12]))
 
pari: [g,chi] = znchar(Mod(97,3850))
 

Basic properties

Modulus: \(3850\)
Conductor: \(1925\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1925}(97,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3850.db

\(\chi_{3850}(97,\cdot)\) \(\chi_{3850}(223,\cdot)\) \(\chi_{3850}(377,\cdot)\) \(\chi_{3850}(587,\cdot)\) \(\chi_{3850}(1567,\cdot)\) \(\chi_{3850}(2183,\cdot)\) \(\chi_{3850}(2253,\cdot)\) \(\chi_{3850}(3513,\cdot)\)

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: Number field defined by a degree 20 polynomial

Values on generators

\((2927,2201,1751)\) → \((e\left(\frac{17}{20}\right),-1,e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\( \chi_{ 3850 }(97, a) \) \(1\)\(1\)\(i\)\(-1\)\(i\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(-i\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{17}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3850 }(97,a) \;\) at \(\;a = \) e.g. 2