Properties

Label 5445.208
Modulus $5445$
Conductor $605$
Order $44$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5445, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,33,42]))
 
pari: [g,chi] = znchar(Mod(208,5445))
 

Basic properties

Modulus: \(5445\)
Conductor: \(605\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{605}(208,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5445.cd

\(\chi_{5445}(208,\cdot)\) \(\chi_{5445}(307,\cdot)\) \(\chi_{5445}(703,\cdot)\) \(\chi_{5445}(802,\cdot)\) \(\chi_{5445}(1198,\cdot)\) \(\chi_{5445}(1297,\cdot)\) \(\chi_{5445}(1792,\cdot)\) \(\chi_{5445}(2188,\cdot)\) \(\chi_{5445}(2287,\cdot)\) \(\chi_{5445}(2683,\cdot)\) \(\chi_{5445}(3178,\cdot)\) \(\chi_{5445}(3277,\cdot)\) \(\chi_{5445}(3673,\cdot)\) \(\chi_{5445}(3772,\cdot)\) \(\chi_{5445}(4168,\cdot)\) \(\chi_{5445}(4267,\cdot)\) \(\chi_{5445}(4663,\cdot)\) \(\chi_{5445}(4762,\cdot)\) \(\chi_{5445}(5158,\cdot)\) \(\chi_{5445}(5257,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: 44.44.2885428559557085084648615903962269104974580506944665166312236845353556846511909399754484184086322784423828125.1

Values on generators

\((3026,4357,3511)\) → \((1,-i,e\left(\frac{21}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(23\)
\( \chi_{ 5445 }(208, a) \) \(1\)\(1\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{3}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5445 }(208,a) \;\) at \(\;a = \) e.g. 2