Properties

Label 4020.2693
Modulus $4020$
Conductor $1005$
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([0,66,99,38]))
 
pari: [g,chi] = znchar(Mod(2693,4020))
 

Basic properties

Modulus: \(4020\)
Conductor: \(1005\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1005}(683,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4020.do

\(\chi_{4020}(113,\cdot)\) \(\chi_{4020}(197,\cdot)\) \(\chi_{4020}(233,\cdot)\) \(\chi_{4020}(353,\cdot)\) \(\chi_{4020}(413,\cdot)\) \(\chi_{4020}(497,\cdot)\) \(\chi_{4020}(593,\cdot)\) \(\chi_{4020}(653,\cdot)\) \(\chi_{4020}(677,\cdot)\) \(\chi_{4020}(917,\cdot)\) \(\chi_{4020}(1037,\cdot)\) \(\chi_{4020}(1133,\cdot)\) \(\chi_{4020}(1157,\cdot)\) \(\chi_{4020}(1217,\cdot)\) \(\chi_{4020}(1397,\cdot)\) \(\chi_{4020}(1457,\cdot)\) \(\chi_{4020}(1553,\cdot)\) \(\chi_{4020}(1793,\cdot)\) \(\chi_{4020}(1853,\cdot)\) \(\chi_{4020}(1937,\cdot)\) \(\chi_{4020}(2213,\cdot)\) \(\chi_{4020}(2357,\cdot)\) \(\chi_{4020}(2393,\cdot)\) \(\chi_{4020}(2453,\cdot)\) \(\chi_{4020}(2513,\cdot)\) \(\chi_{4020}(2597,\cdot)\) \(\chi_{4020}(2633,\cdot)\) \(\chi_{4020}(2657,\cdot)\) \(\chi_{4020}(2693,\cdot)\) \(\chi_{4020}(3017,\cdot)\) ...

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

\((2011,2681,3217,1141)\) → \((1,-1,-i,e\left(\frac{19}{66}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 4020 }(2693, a) \) \(-1\)\(1\)\(e\left(\frac{49}{132}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{95}{132}\right)\)\(e\left(\frac{89}{132}\right)\)\(e\left(\frac{25}{66}\right)\)\(e\left(\frac{107}{132}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{25}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4020 }(2693,a) \;\) at \(\;a = \) e.g. 2