Properties

Label 1620.29
Modulus $1620$
Conductor $405$
Order $54$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(54))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,37,27]))
 
pari: [g,chi] = znchar(Mod(29,1620))
 

Basic properties

Modulus: \(1620\)
Conductor: \(405\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{405}(29,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1620.bl

\(\chi_{1620}(29,\cdot)\) \(\chi_{1620}(149,\cdot)\) \(\chi_{1620}(209,\cdot)\) \(\chi_{1620}(329,\cdot)\) \(\chi_{1620}(389,\cdot)\) \(\chi_{1620}(509,\cdot)\) \(\chi_{1620}(569,\cdot)\) \(\chi_{1620}(689,\cdot)\) \(\chi_{1620}(749,\cdot)\) \(\chi_{1620}(869,\cdot)\) \(\chi_{1620}(929,\cdot)\) \(\chi_{1620}(1049,\cdot)\) \(\chi_{1620}(1109,\cdot)\) \(\chi_{1620}(1229,\cdot)\) \(\chi_{1620}(1289,\cdot)\) \(\chi_{1620}(1409,\cdot)\) \(\chi_{1620}(1469,\cdot)\) \(\chi_{1620}(1589,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 54 polynomial

Values on generators

\((811,1541,1297)\) → \((1,e\left(\frac{37}{54}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1620 }(29, a) \) \(-1\)\(1\)\(e\left(\frac{25}{54}\right)\)\(e\left(\frac{49}{54}\right)\)\(e\left(\frac{53}{54}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{27}\right)\)\(e\left(\frac{19}{54}\right)\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{17}{54}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1620 }(29,a) \;\) at \(\;a = \) e.g. 2