Properties

Label 4002.17
Modulus $4002$
Conductor $2001$
Order $44$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,14,33]))
 
pari: [g,chi] = znchar(Mod(17,4002))
 

Basic properties

Modulus: \(4002\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2001}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4002.bj

\(\chi_{4002}(17,\cdot)\) \(\chi_{4002}(191,\cdot)\) \(\chi_{4002}(365,\cdot)\) \(\chi_{4002}(389,\cdot)\) \(\chi_{4002}(539,\cdot)\) \(\chi_{4002}(563,\cdot)\) \(\chi_{4002}(911,\cdot)\) \(\chi_{4002}(1259,\cdot)\) \(\chi_{4002}(1433,\cdot)\) \(\chi_{4002}(1583,\cdot)\) \(\chi_{4002}(1607,\cdot)\) \(\chi_{4002}(1781,\cdot)\) \(\chi_{4002}(2453,\cdot)\) \(\chi_{4002}(2627,\cdot)\) \(\chi_{4002}(2825,\cdot)\) \(\chi_{4002}(3149,\cdot)\) \(\chi_{4002}(3323,\cdot)\) \(\chi_{4002}(3671,\cdot)\) \(\chi_{4002}(3695,\cdot)\) \(\chi_{4002}(3869,\cdot)\)

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

\((2669,3133,553)\) → \((-1,e\left(\frac{7}{22}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(31\)\(35\)\(37\)
\( \chi_{ 4002 }(17, a) \) \(-1\)\(1\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{41}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4002 }(17,a) \;\) at \(\;a = \) e.g. 2