Properties

Label 9075.421
Modulus $9075$
Conductor $3025$
Order $55$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9075, base_ring=CyclotomicField(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,66,18]))
 
pari: [g,chi] = znchar(Mod(421,9075))
 

Basic properties

Modulus: \(9075\)
Conductor: \(3025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(55\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3025}(421,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9075.dv

\(\chi_{9075}(421,\cdot)\) \(\chi_{9075}(631,\cdot)\) \(\chi_{9075}(691,\cdot)\) \(\chi_{9075}(1246,\cdot)\) \(\chi_{9075}(1336,\cdot)\) \(\chi_{9075}(1456,\cdot)\) \(\chi_{9075}(1516,\cdot)\) \(\chi_{9075}(2071,\cdot)\) \(\chi_{9075}(2161,\cdot)\) \(\chi_{9075}(2281,\cdot)\) \(\chi_{9075}(2341,\cdot)\) \(\chi_{9075}(2896,\cdot)\) \(\chi_{9075}(2986,\cdot)\) \(\chi_{9075}(3166,\cdot)\) \(\chi_{9075}(3721,\cdot)\) \(\chi_{9075}(3811,\cdot)\) \(\chi_{9075}(3931,\cdot)\) \(\chi_{9075}(3991,\cdot)\) \(\chi_{9075}(4546,\cdot)\) \(\chi_{9075}(4636,\cdot)\) \(\chi_{9075}(4756,\cdot)\) \(\chi_{9075}(4816,\cdot)\) \(\chi_{9075}(5371,\cdot)\) \(\chi_{9075}(5461,\cdot)\) \(\chi_{9075}(5581,\cdot)\) \(\chi_{9075}(5641,\cdot)\) \(\chi_{9075}(6196,\cdot)\) \(\chi_{9075}(6286,\cdot)\) \(\chi_{9075}(6406,\cdot)\) \(\chi_{9075}(6466,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 55 polynomial

Values on generators

\((3026,727,5326)\) → \((1,e\left(\frac{3}{5}\right),e\left(\frac{9}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(23\)
\( \chi_{ 9075 }(421, a) \) \(1\)\(1\)\(e\left(\frac{42}{55}\right)\)\(e\left(\frac{29}{55}\right)\)\(e\left(\frac{8}{55}\right)\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{51}{55}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{3}{55}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{21}{55}\right)\)\(e\left(\frac{3}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9075 }(421,a) \;\) at \(\;a = \) e.g. 2