Properties

Label 3525.26
Modulus $3525$
Conductor $141$
Order $46$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3525, base_ring=CyclotomicField(46))
 
M = H._module
 
chi = DirichletCharacter(H, M([23,0,29]))
 
pari: [g,chi] = znchar(Mod(26,3525))
 

Basic properties

Modulus: \(3525\)
Conductor: \(141\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(46\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{141}(26,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3525.bc

\(\chi_{3525}(26,\cdot)\) \(\chi_{3525}(176,\cdot)\) \(\chi_{3525}(326,\cdot)\) \(\chi_{3525}(626,\cdot)\) \(\chi_{3525}(701,\cdot)\) \(\chi_{3525}(851,\cdot)\) \(\chi_{3525}(926,\cdot)\) \(\chi_{3525}(1151,\cdot)\) \(\chi_{3525}(1376,\cdot)\) \(\chi_{3525}(1451,\cdot)\) \(\chi_{3525}(1526,\cdot)\) \(\chi_{3525}(1676,\cdot)\) \(\chi_{3525}(1826,\cdot)\) \(\chi_{3525}(2051,\cdot)\) \(\chi_{3525}(2126,\cdot)\) \(\chi_{3525}(2201,\cdot)\) \(\chi_{3525}(2276,\cdot)\) \(\chi_{3525}(2426,\cdot)\) \(\chi_{3525}(2501,\cdot)\) \(\chi_{3525}(2576,\cdot)\) \(\chi_{3525}(2651,\cdot)\) \(\chi_{3525}(3476,\cdot)\)

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: \(\Q(\zeta_{23})\)
Fixed field: \(\Q(\zeta_{141})^+\)

Values on generators

\((2351,1552,2026)\) → \((-1,1,e\left(\frac{29}{46}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 3525 }(26, a) \) \(1\)\(1\)\(e\left(\frac{39}{46}\right)\)\(e\left(\frac{16}{23}\right)\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{25}{46}\right)\)\(e\left(\frac{21}{23}\right)\)\(e\left(\frac{43}{46}\right)\)\(e\left(\frac{1}{46}\right)\)\(e\left(\frac{9}{23}\right)\)\(e\left(\frac{27}{46}\right)\)\(e\left(\frac{17}{46}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3525 }(26,a) \;\) at \(\;a = \) e.g. 2