Properties

Label 4020.2707
Modulus $4020$
Conductor $1340$
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(4020, base_ring=CyclotomicField(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,0,11,34]))
 
pari: [g,chi] = znchar(Mod(2707,4020))
 

Basic properties

Modulus: \(4020\)
Conductor: \(1340\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1340}(27,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4020.cn

\(\chi_{4020}(43,\cdot)\) \(\chi_{4020}(187,\cdot)\) \(\chi_{4020}(343,\cdot)\) \(\chi_{4020}(847,\cdot)\) \(\chi_{4020}(943,\cdot)\) \(\chi_{4020}(1063,\cdot)\) \(\chi_{4020}(1147,\cdot)\) \(\chi_{4020}(1747,\cdot)\) \(\chi_{4020}(1867,\cdot)\) \(\chi_{4020}(1903,\cdot)\) \(\chi_{4020}(2263,\cdot)\) \(\chi_{4020}(2323,\cdot)\) \(\chi_{4020}(2683,\cdot)\) \(\chi_{4020}(2707,\cdot)\) \(\chi_{4020}(2923,\cdot)\) \(\chi_{4020}(3067,\cdot)\) \(\chi_{4020}(3127,\cdot)\) \(\chi_{4020}(3403,\cdot)\) \(\chi_{4020}(3487,\cdot)\) \(\chi_{4020}(3727,\cdot)\)

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

\((2011,2681,3217,1141)\) → \((-1,1,i,e\left(\frac{17}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 4020 }(2707, a) \) \(-1\)\(1\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{39}{44}\right)\)\(-1\)\(e\left(\frac{9}{11}\right)\)\(i\)\(e\left(\frac{21}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4020 }(2707,a) \;\) at \(\;a = \) e.g. 2