Properties

Label 3025.18
Modulus $3025$
Conductor $605$
Order $220$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3025, base_ring=CyclotomicField(220))
 
M = H._module
 
chi = DirichletCharacter(H, M([165,134]))
 
pari: [g,chi] = znchar(Mod(18,3025))
 

Basic properties

Modulus: \(3025\)
Conductor: \(605\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(220\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{605}(18,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3025.cy

\(\chi_{3025}(7,\cdot)\) \(\chi_{3025}(18,\cdot)\) \(\chi_{3025}(57,\cdot)\) \(\chi_{3025}(68,\cdot)\) \(\chi_{3025}(107,\cdot)\) \(\chi_{3025}(182,\cdot)\) \(\chi_{3025}(193,\cdot)\) \(\chi_{3025}(293,\cdot)\) \(\chi_{3025}(332,\cdot)\) \(\chi_{3025}(343,\cdot)\) \(\chi_{3025}(382,\cdot)\) \(\chi_{3025}(393,\cdot)\) \(\chi_{3025}(468,\cdot)\) \(\chi_{3025}(557,\cdot)\) \(\chi_{3025}(568,\cdot)\) \(\chi_{3025}(607,\cdot)\) \(\chi_{3025}(618,\cdot)\) \(\chi_{3025}(657,\cdot)\) \(\chi_{3025}(668,\cdot)\) \(\chi_{3025}(732,\cdot)\) \(\chi_{3025}(743,\cdot)\) \(\chi_{3025}(832,\cdot)\) \(\chi_{3025}(843,\cdot)\) \(\chi_{3025}(882,\cdot)\) \(\chi_{3025}(893,\cdot)\) \(\chi_{3025}(932,\cdot)\) \(\chi_{3025}(943,\cdot)\) \(\chi_{3025}(1007,\cdot)\) \(\chi_{3025}(1018,\cdot)\) \(\chi_{3025}(1107,\cdot)\) ...

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: $\Q(\zeta_{220})$
Fixed field: Number field defined by a degree 220 polynomial (not computed)

Values on generators

\((727,2301)\) → \((-i,e\left(\frac{67}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 3025 }(18, a) \) \(1\)\(1\)\(e\left(\frac{79}{220}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{79}{110}\right)\)\(e\left(\frac{23}{110}\right)\)\(e\left(\frac{3}{220}\right)\)\(e\left(\frac{17}{220}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{169}{220}\right)\)\(e\left(\frac{41}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3025 }(18,a) \;\) at \(\;a = \) e.g. 2