Properties

Label 3525.2026
Modulus $3525$
Conductor $47$
Order $46$
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(46))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,1]))
 
pari: [g,chi] = znchar(Mod(2026,3525))
 

Basic properties

Modulus: \(3525\)
Conductor: \(47\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(46\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{47}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3525.z

\(\chi_{3525}(76,\cdot)\) \(\chi_{3525}(151,\cdot)\) \(\chi_{3525}(226,\cdot)\) \(\chi_{3525}(301,\cdot)\) \(\chi_{3525}(1126,\cdot)\) \(\chi_{3525}(1201,\cdot)\) \(\chi_{3525}(1351,\cdot)\) \(\chi_{3525}(1501,\cdot)\) \(\chi_{3525}(1801,\cdot)\) \(\chi_{3525}(1876,\cdot)\) \(\chi_{3525}(2026,\cdot)\) \(\chi_{3525}(2101,\cdot)\) \(\chi_{3525}(2326,\cdot)\) \(\chi_{3525}(2551,\cdot)\) \(\chi_{3525}(2626,\cdot)\) \(\chi_{3525}(2701,\cdot)\) \(\chi_{3525}(2851,\cdot)\) \(\chi_{3525}(3001,\cdot)\) \(\chi_{3525}(3226,\cdot)\) \(\chi_{3525}(3301,\cdot)\) \(\chi_{3525}(3376,\cdot)\) \(\chi_{3525}(3451,\cdot)\)

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: Number field defined by a degree 46 polynomial

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 3525 }(2026, a) \) \(-1\)\(1\)\(e\left(\frac{9}{23}\right)\)\(e\left(\frac{18}{23}\right)\)\(e\left(\frac{16}{23}\right)\)\(e\left(\frac{4}{23}\right)\)\(e\left(\frac{7}{46}\right)\)\(e\left(\frac{11}{46}\right)\)\(e\left(\frac{2}{23}\right)\)\(e\left(\frac{13}{23}\right)\)\(e\left(\frac{8}{23}\right)\)\(e\left(\frac{45}{46}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3525 }(2026,a) \;\) at \(\;a = \) e.g. 2