Properties

Label 1716.85
Modulus $1716$
Conductor $143$
Order $60$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1716, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,18,55]))
 
pari: [g,chi] = znchar(Mod(85,1716))
 

Basic properties

Modulus: \(1716\)
Conductor: \(143\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{143}(85,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1716.dq

\(\chi_{1716}(85,\cdot)\) \(\chi_{1716}(145,\cdot)\) \(\chi_{1716}(193,\cdot)\) \(\chi_{1716}(349,\cdot)\) \(\chi_{1716}(409,\cdot)\) \(\chi_{1716}(457,\cdot)\) \(\chi_{1716}(613,\cdot)\) \(\chi_{1716}(721,\cdot)\) \(\chi_{1716}(865,\cdot)\) \(\chi_{1716}(877,\cdot)\) \(\chi_{1716}(1129,\cdot)\) \(\chi_{1716}(1333,\cdot)\) \(\chi_{1716}(1393,\cdot)\) \(\chi_{1716}(1597,\cdot)\) \(\chi_{1716}(1645,\cdot)\) \(\chi_{1716}(1657,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((859,1145,937,925)\) → \((1,1,e\left(\frac{3}{10}\right),e\left(\frac{11}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)\(37\)
\( \chi_{ 1716 }(85, a) \) \(1\)\(1\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1716 }(85,a) \;\) at \(\;a = \) e.g. 2