Properties

Label 2009.44
Modulus $2009$
Conductor $2009$
Order $168$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2009, base_ring=CyclotomicField(168))
 
M = H._module
 
chi = DirichletCharacter(H, M([32,63]))
 
pari: [g,chi] = znchar(Mod(44,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(2009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(168\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2009.cc

\(\chi_{2009}(44,\cdot)\) \(\chi_{2009}(109,\cdot)\) \(\chi_{2009}(137,\cdot)\) \(\chi_{2009}(191,\cdot)\) \(\chi_{2009}(219,\cdot)\) \(\chi_{2009}(249,\cdot)\) \(\chi_{2009}(284,\cdot)\) \(\chi_{2009}(331,\cdot)\) \(\chi_{2009}(366,\cdot)\) \(\chi_{2009}(396,\cdot)\) \(\chi_{2009}(424,\cdot)\) \(\chi_{2009}(478,\cdot)\) \(\chi_{2009}(506,\cdot)\) \(\chi_{2009}(536,\cdot)\) \(\chi_{2009}(571,\cdot)\) \(\chi_{2009}(653,\cdot)\) \(\chi_{2009}(683,\cdot)\) \(\chi_{2009}(711,\cdot)\) \(\chi_{2009}(793,\cdot)\) \(\chi_{2009}(823,\cdot)\) \(\chi_{2009}(858,\cdot)\) \(\chi_{2009}(905,\cdot)\) \(\chi_{2009}(940,\cdot)\) \(\chi_{2009}(970,\cdot)\) \(\chi_{2009}(1052,\cdot)\) \(\chi_{2009}(1080,\cdot)\) \(\chi_{2009}(1110,\cdot)\) \(\chi_{2009}(1192,\cdot)\) \(\chi_{2009}(1227,\cdot)\) \(\chi_{2009}(1257,\cdot)\) ...

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: $\Q(\zeta_{168})$
Fixed field: Number field defined by a degree 168 polynomial (not computed)

Values on generators

\((493,785)\) → \((e\left(\frac{4}{21}\right),e\left(\frac{3}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 2009 }(44, a) \) \(-1\)\(1\)\(e\left(\frac{59}{84}\right)\)\(e\left(\frac{137}{168}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{65}{84}\right)\)\(e\left(\frac{29}{56}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{53}{84}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{125}{168}\right)\)\(e\left(\frac{37}{168}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2009 }(44,a) \;\) at \(\;a = \) e.g. 2