Properties

Label 6040.49
Modulus $6040$
Conductor $755$
Order $150$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6040, base_ring=CyclotomicField(150))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,75,134]))
 
pari: [g,chi] = znchar(Mod(49,6040))
 

Basic properties

Modulus: \(6040\)
Conductor: \(755\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(150\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{755}(49,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6040.et

\(\chi_{6040}(49,\cdot)\) \(\chi_{6040}(169,\cdot)\) \(\chi_{6040}(209,\cdot)\) \(\chi_{6040}(289,\cdot)\) \(\chi_{6040}(489,\cdot)\) \(\chi_{6040}(569,\cdot)\) \(\chi_{6040}(609,\cdot)\) \(\chi_{6040}(649,\cdot)\) \(\chi_{6040}(1009,\cdot)\) \(\chi_{6040}(1329,\cdot)\) \(\chi_{6040}(1369,\cdot)\) \(\chi_{6040}(1449,\cdot)\) \(\chi_{6040}(1609,\cdot)\) \(\chi_{6040}(1649,\cdot)\) \(\chi_{6040}(1849,\cdot)\) \(\chi_{6040}(2169,\cdot)\) \(\chi_{6040}(2209,\cdot)\) \(\chi_{6040}(2409,\cdot)\) \(\chi_{6040}(2609,\cdot)\) \(\chi_{6040}(2729,\cdot)\) \(\chi_{6040}(2969,\cdot)\) \(\chi_{6040}(3089,\cdot)\) \(\chi_{6040}(3369,\cdot)\) \(\chi_{6040}(3609,\cdot)\) \(\chi_{6040}(3649,\cdot)\) \(\chi_{6040}(3769,\cdot)\) \(\chi_{6040}(3809,\cdot)\) \(\chi_{6040}(3849,\cdot)\) \(\chi_{6040}(3969,\cdot)\) \(\chi_{6040}(4249,\cdot)\) ...

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: $\Q(\zeta_{75})$
Fixed field: Number field defined by a degree 150 polynomial (not computed)

Values on generators

\((1511,3021,2417,761)\) → \((1,1,-1,e\left(\frac{67}{75}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 6040 }(49, a) \) \(1\)\(1\)\(e\left(\frac{43}{50}\right)\)\(e\left(\frac{53}{150}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{73}{75}\right)\)\(e\left(\frac{19}{150}\right)\)\(e\left(\frac{47}{150}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{16}{75}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{29}{50}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6040 }(49,a) \;\) at \(\;a = \) e.g. 2