Properties

Label 5025.371
Modulus $5025$
Conductor $5025$
Order $330$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5025, base_ring=CyclotomicField(330))
 
M = H._module
 
chi = DirichletCharacter(H, M([165,198,70]))
 
pari: [g,chi] = znchar(Mod(371,5025))
 

Basic properties

Modulus: \(5025\)
Conductor: \(5025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5025.dn

\(\chi_{5025}(56,\cdot)\) \(\chi_{5025}(71,\cdot)\) \(\chi_{5025}(86,\cdot)\) \(\chi_{5025}(116,\cdot)\) \(\chi_{5025}(236,\cdot)\) \(\chi_{5025}(266,\cdot)\) \(\chi_{5025}(341,\cdot)\) \(\chi_{5025}(356,\cdot)\) \(\chi_{5025}(371,\cdot)\) \(\chi_{5025}(596,\cdot)\) \(\chi_{5025}(686,\cdot)\) \(\chi_{5025}(791,\cdot)\) \(\chi_{5025}(821,\cdot)\) \(\chi_{5025}(881,\cdot)\) \(\chi_{5025}(971,\cdot)\) \(\chi_{5025}(1031,\cdot)\) \(\chi_{5025}(1061,\cdot)\) \(\chi_{5025}(1091,\cdot)\) \(\chi_{5025}(1121,\cdot)\) \(\chi_{5025}(1241,\cdot)\) \(\chi_{5025}(1271,\cdot)\) \(\chi_{5025}(1346,\cdot)\) \(\chi_{5025}(1361,\cdot)\) \(\chi_{5025}(1631,\cdot)\) \(\chi_{5025}(1691,\cdot)\) \(\chi_{5025}(1781,\cdot)\) \(\chi_{5025}(1796,\cdot)\) \(\chi_{5025}(1856,\cdot)\) \(\chi_{5025}(1886,\cdot)\) \(\chi_{5025}(1931,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((1676,202,3151)\) → \((-1,e\left(\frac{3}{5}\right),e\left(\frac{7}{33}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 5025 }(371, a) \) \(-1\)\(1\)\(e\left(\frac{103}{330}\right)\)\(e\left(\frac{103}{165}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{103}{110}\right)\)\(e\left(\frac{203}{330}\right)\)\(e\left(\frac{71}{165}\right)\)\(e\left(\frac{21}{110}\right)\)\(e\left(\frac{41}{165}\right)\)\(e\left(\frac{289}{330}\right)\)\(e\left(\frac{152}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5025 }(371,a) \;\) at \(\;a = \) e.g. 2