Properties

Label 2450.61
Modulus $2450$
Conductor $1225$
Order $210$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2450, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([168,55]))
 
pari: [g,chi] = znchar(Mod(61,2450))
 

Basic properties

Modulus: \(2450\)
Conductor: \(1225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1225}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2450.bs

\(\chi_{2450}(61,\cdot)\) \(\chi_{2450}(131,\cdot)\) \(\chi_{2450}(171,\cdot)\) \(\chi_{2450}(241,\cdot)\) \(\chi_{2450}(271,\cdot)\) \(\chi_{2450}(311,\cdot)\) \(\chi_{2450}(341,\cdot)\) \(\chi_{2450}(381,\cdot)\) \(\chi_{2450}(481,\cdot)\) \(\chi_{2450}(591,\cdot)\) \(\chi_{2450}(621,\cdot)\) \(\chi_{2450}(661,\cdot)\) \(\chi_{2450}(691,\cdot)\) \(\chi_{2450}(731,\cdot)\) \(\chi_{2450}(761,\cdot)\) \(\chi_{2450}(831,\cdot)\) \(\chi_{2450}(871,\cdot)\) \(\chi_{2450}(941,\cdot)\) \(\chi_{2450}(971,\cdot)\) \(\chi_{2450}(1041,\cdot)\) \(\chi_{2450}(1081,\cdot)\) \(\chi_{2450}(1111,\cdot)\) \(\chi_{2450}(1181,\cdot)\) \(\chi_{2450}(1221,\cdot)\) \(\chi_{2450}(1291,\cdot)\) \(\chi_{2450}(1321,\cdot)\) \(\chi_{2450}(1361,\cdot)\) \(\chi_{2450}(1431,\cdot)\) \(\chi_{2450}(1461,\cdot)\) \(\chi_{2450}(1531,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial (not computed)

Values on generators

\((1177,101)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{11}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 2450 }(61, a) \) \(-1\)\(1\)\(e\left(\frac{181}{210}\right)\)\(e\left(\frac{76}{105}\right)\)\(e\left(\frac{29}{105}\right)\)\(e\left(\frac{59}{70}\right)\)\(e\left(\frac{199}{210}\right)\)\(e\left(\frac{17}{30}\right)\)\(e\left(\frac{79}{105}\right)\)\(e\left(\frac{41}{70}\right)\)\(e\left(\frac{11}{35}\right)\)\(e\left(\frac{7}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2450 }(61,a) \;\) at \(\;a = \) e.g. 2