Properties

Label 4017.1756
Modulus $4017$
Conductor $103$
Order $102$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4017, base_ring=CyclotomicField(102))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,1]))
 
pari: [g,chi] = znchar(Mod(1756,4017))
 

Basic properties

Modulus: \(4017\)
Conductor: \(103\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(102\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{103}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4017.dr

\(\chi_{4017}(40,\cdot)\) \(\chi_{4017}(157,\cdot)\) \(\chi_{4017}(352,\cdot)\) \(\chi_{4017}(508,\cdot)\) \(\chi_{4017}(586,\cdot)\) \(\chi_{4017}(703,\cdot)\) \(\chi_{4017}(742,\cdot)\) \(\chi_{4017}(820,\cdot)\) \(\chi_{4017}(859,\cdot)\) \(\chi_{4017}(898,\cdot)\) \(\chi_{4017}(1015,\cdot)\) \(\chi_{4017}(1210,\cdot)\) \(\chi_{4017}(1756,\cdot)\) \(\chi_{4017}(1795,\cdot)\) \(\chi_{4017}(2146,\cdot)\) \(\chi_{4017}(2341,\cdot)\) \(\chi_{4017}(2380,\cdot)\) \(\chi_{4017}(2653,\cdot)\) \(\chi_{4017}(2731,\cdot)\) \(\chi_{4017}(2848,\cdot)\) \(\chi_{4017}(3160,\cdot)\) \(\chi_{4017}(3199,\cdot)\) \(\chi_{4017}(3238,\cdot)\) \(\chi_{4017}(3277,\cdot)\) \(\chi_{4017}(3316,\cdot)\) \(\chi_{4017}(3550,\cdot)\) \(\chi_{4017}(3589,\cdot)\) \(\chi_{4017}(3667,\cdot)\) \(\chi_{4017}(3706,\cdot)\) \(\chi_{4017}(3823,\cdot)\) ...

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: $\Q(\zeta_{51})$
Fixed field: Number field defined by a degree 102 polynomial (not computed)

Values on generators

\((1340,1237,1756)\) → \((1,1,e\left(\frac{1}{102}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(14\)\(16\)\(17\)
\( \chi_{ 4017 }(1756, a) \) \(-1\)\(1\)\(e\left(\frac{22}{51}\right)\)\(e\left(\frac{44}{51}\right)\)\(e\left(\frac{1}{102}\right)\)\(e\left(\frac{2}{51}\right)\)\(e\left(\frac{5}{17}\right)\)\(e\left(\frac{15}{34}\right)\)\(e\left(\frac{61}{102}\right)\)\(e\left(\frac{8}{17}\right)\)\(e\left(\frac{37}{51}\right)\)\(e\left(\frac{35}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4017 }(1756,a) \;\) at \(\;a = \) e.g. 2