Properties

Label 3800.243
Modulus $3800$
Conductor $760$
Order $36$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3800, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([18,18,27,22]))
 
pari: [g,chi] = znchar(Mod(243,3800))
 

Basic properties

Modulus: \(3800\)
Conductor: \(760\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{760}(243,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3800.eb

\(\chi_{3800}(243,\cdot)\) \(\chi_{3800}(307,\cdot)\) \(\chi_{3800}(507,\cdot)\) \(\chi_{3800}(907,\cdot)\) \(\chi_{3800}(1307,\cdot)\) \(\chi_{3800}(2043,\cdot)\) \(\chi_{3800}(2643,\cdot)\) \(\chi_{3800}(3043,\cdot)\) \(\chi_{3800}(3107,\cdot)\) \(\chi_{3800}(3243,\cdot)\) \(\chi_{3800}(3643,\cdot)\) \(\chi_{3800}(3707,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.0.4031181156993454136731178943694064571490658196389888000000000000000000000000000.1

Values on generators

\((951,1901,1977,401)\) → \((-1,-1,-i,e\left(\frac{11}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 3800 }(243, a) \) \(-1\)\(1\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{7}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3800 }(243,a) \;\) at \(\;a = \) e.g. 2