Properties

Label 9680.23
Modulus $9680$
Conductor $4840$
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(9680, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,22,33,28]))
 
pari: [g,chi] = znchar(Mod(23,9680))
 

Basic properties

Modulus: \(9680\)
Conductor: \(4840\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4840}(2443,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9680.eg

\(\chi_{9680}(23,\cdot)\) \(\chi_{9680}(903,\cdot)\) \(\chi_{9680}(1607,\cdot)\) \(\chi_{9680}(1783,\cdot)\) \(\chi_{9680}(2487,\cdot)\) \(\chi_{9680}(3367,\cdot)\) \(\chi_{9680}(3543,\cdot)\) \(\chi_{9680}(4247,\cdot)\) \(\chi_{9680}(4423,\cdot)\) \(\chi_{9680}(5127,\cdot)\) \(\chi_{9680}(5303,\cdot)\) \(\chi_{9680}(6007,\cdot)\) \(\chi_{9680}(6183,\cdot)\) \(\chi_{9680}(6887,\cdot)\) \(\chi_{9680}(7063,\cdot)\) \(\chi_{9680}(7767,\cdot)\) \(\chi_{9680}(7943,\cdot)\) \(\chi_{9680}(8647,\cdot)\) \(\chi_{9680}(8823,\cdot)\) \(\chi_{9680}(9527,\cdot)\)

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: Number field defined by a degree 44 polynomial

Values on generators

\((3631,2421,1937,4721)\) → \((-1,-1,-i,e\left(\frac{7}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 9680 }(23, a) \) \(1\)\(1\)\(i\)\(e\left(\frac{31}{44}\right)\)\(-1\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{13}{44}\right)\)\(-i\)\(e\left(\frac{9}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9680 }(23,a) \;\) at \(\;a = \) e.g. 2