Properties

Label 1035.61
Modulus $1035$
Conductor $207$
Order $66$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1035, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,0,51]))
 
pari: [g,chi] = znchar(Mod(61,1035))
 

Basic properties

Modulus: \(1035\)
Conductor: \(207\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{207}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1035.bq

\(\chi_{1035}(61,\cdot)\) \(\chi_{1035}(76,\cdot)\) \(\chi_{1035}(106,\cdot)\) \(\chi_{1035}(166,\cdot)\) \(\chi_{1035}(241,\cdot)\) \(\chi_{1035}(286,\cdot)\) \(\chi_{1035}(421,\cdot)\) \(\chi_{1035}(481,\cdot)\) \(\chi_{1035}(511,\cdot)\) \(\chi_{1035}(526,\cdot)\) \(\chi_{1035}(571,\cdot)\) \(\chi_{1035}(661,\cdot)\) \(\chi_{1035}(751,\cdot)\) \(\chi_{1035}(796,\cdot)\) \(\chi_{1035}(826,\cdot)\) \(\chi_{1035}(871,\cdot)\) \(\chi_{1035}(916,\cdot)\) \(\chi_{1035}(931,\cdot)\) \(\chi_{1035}(976,\cdot)\) \(\chi_{1035}(1006,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((461,622,856)\) → \((e\left(\frac{2}{3}\right),1,e\left(\frac{17}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 1035 }(61, a) \) \(-1\)\(1\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{23}{66}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{41}{66}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{37}{66}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{13}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1035 }(61,a) \;\) at \(\;a = \) e.g. 2