Properties

Label 1035.32
Modulus $1035$
Conductor $1035$
Order $132$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1035, base_ring=CyclotomicField(132))
 
M = H._module
 
chi = DirichletCharacter(H, M([110,33,60]))
 
pari: [g,chi] = znchar(Mod(32,1035))
 

Basic properties

Modulus: \(1035\)
Conductor: \(1035\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1035.bt

\(\chi_{1035}(2,\cdot)\) \(\chi_{1035}(32,\cdot)\) \(\chi_{1035}(77,\cdot)\) \(\chi_{1035}(128,\cdot)\) \(\chi_{1035}(167,\cdot)\) \(\chi_{1035}(173,\cdot)\) \(\chi_{1035}(248,\cdot)\) \(\chi_{1035}(257,\cdot)\) \(\chi_{1035}(302,\cdot)\) \(\chi_{1035}(308,\cdot)\) \(\chi_{1035}(317,\cdot)\) \(\chi_{1035}(338,\cdot)\) \(\chi_{1035}(347,\cdot)\) \(\chi_{1035}(353,\cdot)\) \(\chi_{1035}(407,\cdot)\) \(\chi_{1035}(443,\cdot)\) \(\chi_{1035}(473,\cdot)\) \(\chi_{1035}(518,\cdot)\) \(\chi_{1035}(533,\cdot)\) \(\chi_{1035}(542,\cdot)\) \(\chi_{1035}(578,\cdot)\) \(\chi_{1035}(587,\cdot)\) \(\chi_{1035}(623,\cdot)\) \(\chi_{1035}(653,\cdot)\) \(\chi_{1035}(662,\cdot)\) \(\chi_{1035}(698,\cdot)\) \(\chi_{1035}(722,\cdot)\) \(\chi_{1035}(752,\cdot)\) \(\chi_{1035}(767,\cdot)\) \(\chi_{1035}(788,\cdot)\) ...

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: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial (not computed)

Values on generators

\((461,622,856)\) → \((e\left(\frac{5}{6}\right),i,e\left(\frac{5}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 1035 }(32, a) \) \(1\)\(1\)\(e\left(\frac{131}{132}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{29}{132}\right)\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{61}{66}\right)\)\(e\left(\frac{103}{132}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{7}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1035 }(32,a) \;\) at \(\;a = \) e.g. 2