Properties

Label 4020.3011
Modulus $4020$
Conductor $804$
Order $66$
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(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,33,0,35]))
 
pari: [g,chi] = znchar(Mod(3011,4020))
 

Basic properties

Modulus: \(4020\)
Conductor: \(804\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{804}(599,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4020.de

\(\chi_{4020}(11,\cdot)\) \(\chi_{4020}(191,\cdot)\) \(\chi_{4020}(251,\cdot)\) \(\chi_{4020}(731,\cdot)\) \(\chi_{4020}(1151,\cdot)\) \(\chi_{4020}(1391,\cdot)\) \(\chi_{4020}(1451,\cdot)\) \(\chi_{4020}(1811,\cdot)\) \(\chi_{4020}(1991,\cdot)\) \(\chi_{4020}(2051,\cdot)\) \(\chi_{4020}(2111,\cdot)\) \(\chi_{4020}(2231,\cdot)\) \(\chi_{4020}(2291,\cdot)\) \(\chi_{4020}(2711,\cdot)\) \(\chi_{4020}(3011,\cdot)\) \(\chi_{4020}(3311,\cdot)\) \(\chi_{4020}(3491,\cdot)\) \(\chi_{4020}(3731,\cdot)\) \(\chi_{4020}(3851,\cdot)\) \(\chi_{4020}(3971,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((2011,2681,3217,1141)\) → \((-1,-1,1,e\left(\frac{35}{66}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 4020 }(3011, a) \) \(-1\)\(1\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{5}{66}\right)\)\(e\left(\frac{29}{66}\right)\)\(e\left(\frac{53}{66}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{20}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4020 }(3011,a) \;\) at \(\;a = \) e.g. 2