Properties

Label 4020.1349
Modulus $4020$
Conductor $1005$
Order $22$
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(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,11,11,4]))
 
pari: [g,chi] = znchar(Mod(1349,4020))
 

Basic properties

Modulus: \(4020\)
Conductor: \(1005\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1005}(344,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4020.ck

\(\chi_{4020}(89,\cdot)\) \(\chi_{4020}(149,\cdot)\) \(\chi_{4020}(509,\cdot)\) \(\chi_{4020}(1349,\cdot)\) \(\chi_{4020}(1469,\cdot)\) \(\chi_{4020}(2069,\cdot)\) \(\chi_{4020}(2369,\cdot)\) \(\chi_{4020}(3029,\cdot)\) \(\chi_{4020}(3509,\cdot)\) \(\chi_{4020}(3749,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: Number field defined by a degree 22 polynomial

Values on generators

\((2011,2681,3217,1141)\) → \((1,-1,-1,e\left(\frac{2}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 4020 }(1349, a) \) \(-1\)\(1\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(-1\)\(e\left(\frac{6}{11}\right)\)\(-1\)\(e\left(\frac{3}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4020 }(1349,a) \;\) at \(\;a = \) e.g. 2