Properties

Label 6840.1117
Modulus $6840$
Conductor $760$
Order $36$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6840, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,18,0,9,22]))
 
pari: [g,chi] = znchar(Mod(1117,6840))
 

Basic properties

Modulus: \(6840\)
Conductor: \(760\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{760}(357,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6840.ol

\(\chi_{6840}(1117,\cdot)\) \(\chi_{6840}(1333,\cdot)\) \(\chi_{6840}(1477,\cdot)\) \(\chi_{6840}(1693,\cdot)\) \(\chi_{6840}(1837,\cdot)\) \(\chi_{6840}(2917,\cdot)\) \(\chi_{6840}(3853,\cdot)\) \(\chi_{6840}(4213,\cdot)\) \(\chi_{6840}(4573,\cdot)\) \(\chi_{6840}(5437,\cdot)\) \(\chi_{6840}(5653,\cdot)\) \(\chi_{6840}(5797,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.36.4031181156993454136731178943694064571490658196389888000000000000000000000000000.1

Values on generators

\((1711,3421,5321,2737,6481)\) → \((1,-1,1,i,e\left(\frac{11}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 6840 }(1117, a) \) \(1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{1}{6}\right)\)\(i\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{1}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6840 }(1117,a) \;\) at \(\;a = \) e.g. 2