Properties

Label 2450.59
Modulus $2450$
Conductor $1225$
Order $210$
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(2450, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([147,65]))
 
pari: [g,chi] = znchar(Mod(59,2450))
 

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 \(\chi_{1225}(59,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2450.br

\(\chi_{2450}(59,\cdot)\) \(\chi_{2450}(89,\cdot)\) \(\chi_{2450}(159,\cdot)\) \(\chi_{2450}(229,\cdot)\) \(\chi_{2450}(269,\cdot)\) \(\chi_{2450}(339,\cdot)\) \(\chi_{2450}(369,\cdot)\) \(\chi_{2450}(409,\cdot)\) \(\chi_{2450}(439,\cdot)\) \(\chi_{2450}(479,\cdot)\) \(\chi_{2450}(579,\cdot)\) \(\chi_{2450}(689,\cdot)\) \(\chi_{2450}(719,\cdot)\) \(\chi_{2450}(759,\cdot)\) \(\chi_{2450}(789,\cdot)\) \(\chi_{2450}(829,\cdot)\) \(\chi_{2450}(859,\cdot)\) \(\chi_{2450}(929,\cdot)\) \(\chi_{2450}(969,\cdot)\) \(\chi_{2450}(1039,\cdot)\) \(\chi_{2450}(1069,\cdot)\) \(\chi_{2450}(1139,\cdot)\) \(\chi_{2450}(1179,\cdot)\) \(\chi_{2450}(1209,\cdot)\) \(\chi_{2450}(1279,\cdot)\) \(\chi_{2450}(1319,\cdot)\) \(\chi_{2450}(1389,\cdot)\) \(\chi_{2450}(1419,\cdot)\) \(\chi_{2450}(1459,\cdot)\) \(\chi_{2450}(1529,\cdot)\) ...

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

\((1177,101)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{13}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 2450 }(59, a) \) \(-1\)\(1\)\(e\left(\frac{22}{105}\right)\)\(e\left(\frac{44}{105}\right)\)\(e\left(\frac{61}{105}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{88}{105}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{97}{210}\right)\)\(e\left(\frac{22}{35}\right)\)\(e\left(\frac{34}{35}\right)\)\(e\left(\frac{23}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2450 }(59,a) \;\) at \(\;a = \) e.g. 2