Properties

Label 3525.1552
Modulus $3525$
Conductor $25$
Order $20$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3525, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,1,0]))
 
pari: [g,chi] = znchar(Mod(1552,3525))
 

Basic properties

Modulus: \(3525\)
Conductor: \(25\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{25}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3525.v

\(\chi_{3525}(142,\cdot)\) \(\chi_{3525}(283,\cdot)\) \(\chi_{3525}(847,\cdot)\) \(\chi_{3525}(988,\cdot)\) \(\chi_{3525}(1552,\cdot)\) \(\chi_{3525}(2398,\cdot)\) \(\chi_{3525}(2962,\cdot)\) \(\chi_{3525}(3103,\cdot)\)

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

\((2351,1552,2026)\) → \((1,e\left(\frac{1}{20}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 3525 }(1552, a) \) \(-1\)\(1\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(i\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{9}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3525 }(1552,a) \;\) at \(\;a = \) e.g. 2