Properties

Label 3450.7
Modulus $3450$
Conductor $115$
Order $44$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3450, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,11,38]))
 
pari: [g,chi] = znchar(Mod(7,3450))
 

Basic properties

Modulus: \(3450\)
Conductor: \(115\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{115}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3450.bi

\(\chi_{3450}(7,\cdot)\) \(\chi_{3450}(43,\cdot)\) \(\chi_{3450}(157,\cdot)\) \(\chi_{3450}(343,\cdot)\) \(\chi_{3450}(457,\cdot)\) \(\chi_{3450}(493,\cdot)\) \(\chi_{3450}(757,\cdot)\) \(\chi_{3450}(793,\cdot)\) \(\chi_{3450}(907,\cdot)\) \(\chi_{3450}(1207,\cdot)\) \(\chi_{3450}(1693,\cdot)\) \(\chi_{3450}(1993,\cdot)\) \(\chi_{3450}(2107,\cdot)\) \(\chi_{3450}(2407,\cdot)\) \(\chi_{3450}(2443,\cdot)\) \(\chi_{3450}(2593,\cdot)\) \(\chi_{3450}(2857,\cdot)\) \(\chi_{3450}(3007,\cdot)\) \(\chi_{3450}(3043,\cdot)\) \(\chi_{3450}(3193,\cdot)\)

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: \(\Q(\zeta_{115})^+\)

Values on generators

\((1151,277,1201)\) → \((1,i,e\left(\frac{19}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3450 }(7, a) \) \(1\)\(1\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{3}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3450 }(7,a) \;\) at \(\;a = \) e.g. 2