Properties

Label 1350.421
Modulus $1350$
Conductor $675$
Order $45$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,54]))
 
pari: [g,chi] = znchar(Mod(421,1350))
 

Basic properties

Modulus: \(1350\)
Conductor: \(675\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(45\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{675}(421,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1350.bc

\(\chi_{1350}(31,\cdot)\) \(\chi_{1350}(61,\cdot)\) \(\chi_{1350}(121,\cdot)\) \(\chi_{1350}(211,\cdot)\) \(\chi_{1350}(241,\cdot)\) \(\chi_{1350}(331,\cdot)\) \(\chi_{1350}(391,\cdot)\) \(\chi_{1350}(421,\cdot)\) \(\chi_{1350}(481,\cdot)\) \(\chi_{1350}(511,\cdot)\) \(\chi_{1350}(571,\cdot)\) \(\chi_{1350}(661,\cdot)\) \(\chi_{1350}(691,\cdot)\) \(\chi_{1350}(781,\cdot)\) \(\chi_{1350}(841,\cdot)\) \(\chi_{1350}(871,\cdot)\) \(\chi_{1350}(931,\cdot)\) \(\chi_{1350}(961,\cdot)\) \(\chi_{1350}(1021,\cdot)\) \(\chi_{1350}(1111,\cdot)\) \(\chi_{1350}(1141,\cdot)\) \(\chi_{1350}(1231,\cdot)\) \(\chi_{1350}(1291,\cdot)\) \(\chi_{1350}(1321,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 45 polynomial

Values on generators

\((1001,1027)\) → \((e\left(\frac{2}{9}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 1350 }(421, a) \) \(1\)\(1\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{8}{45}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{2}{45}\right)\)\(e\left(\frac{19}{45}\right)\)\(e\left(\frac{11}{45}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{8}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1350 }(421,a) \;\) at \(\;a = \) e.g. 2