Properties

Label 4020.853
Modulus $4020$
Conductor $335$
Order $132$
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(132))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,99,92]))
 
pari: [g,chi] = znchar(Mod(853,4020))
 

Basic properties

Modulus: \(4020\)
Conductor: \(335\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{335}(183,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4020.dp

\(\chi_{4020}(73,\cdot)\) \(\chi_{4020}(157,\cdot)\) \(\chi_{4020}(217,\cdot)\) \(\chi_{4020}(457,\cdot)\) \(\chi_{4020}(553,\cdot)\) \(\chi_{4020}(613,\cdot)\) \(\chi_{4020}(793,\cdot)\) \(\chi_{4020}(853,\cdot)\) \(\chi_{4020}(877,\cdot)\) \(\chi_{4020}(973,\cdot)\) \(\chi_{4020}(1093,\cdot)\) \(\chi_{4020}(1333,\cdot)\) \(\chi_{4020}(1357,\cdot)\) \(\chi_{4020}(1417,\cdot)\) \(\chi_{4020}(1513,\cdot)\) \(\chi_{4020}(1597,\cdot)\) \(\chi_{4020}(1657,\cdot)\) \(\chi_{4020}(1777,\cdot)\) \(\chi_{4020}(1813,\cdot)\) \(\chi_{4020}(1897,\cdot)\) \(\chi_{4020}(2113,\cdot)\) \(\chi_{4020}(2137,\cdot)\) \(\chi_{4020}(2317,\cdot)\) \(\chi_{4020}(2533,\cdot)\) \(\chi_{4020}(2593,\cdot)\) \(\chi_{4020}(2617,\cdot)\) \(\chi_{4020}(2713,\cdot)\) \(\chi_{4020}(2773,\cdot)\) \(\chi_{4020}(2833,\cdot)\) \(\chi_{4020}(2917,\cdot)\) ...

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: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial (not computed)

Values on generators

\((2011,2681,3217,1141)\) → \((1,1,-i,e\left(\frac{23}{33}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 4020 }(853, a) \) \(-1\)\(1\)\(e\left(\frac{103}{132}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{65}{132}\right)\)\(e\left(\frac{47}{132}\right)\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{101}{132}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{31}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4020 }(853,a) \;\) at \(\;a = \) e.g. 2