Properties

Label 1800.1361
Modulus $1800$
Conductor $225$
Order $30$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,5,24]))
 
pari: [g,chi] = znchar(Mod(1361,1800))
 

Basic properties

Modulus: \(1800\)
Conductor: \(225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{225}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1800.cy

\(\chi_{1800}(41,\cdot)\) \(\chi_{1800}(281,\cdot)\) \(\chi_{1800}(641,\cdot)\) \(\chi_{1800}(761,\cdot)\) \(\chi_{1800}(1121,\cdot)\) \(\chi_{1800}(1361,\cdot)\) \(\chi_{1800}(1481,\cdot)\) \(\chi_{1800}(1721,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.0.10495827164017277150673379537693108431994915008544921875.1

Values on generators

\((1351,901,1001,577)\) → \((1,1,e\left(\frac{1}{6}\right),e\left(\frac{4}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1800 }(1361, a) \) \(-1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1800 }(1361,a) \;\) at \(\;a = \) e.g. 2