Properties

Label 3025.42
Modulus $3025$
Conductor $3025$
Order $220$
Real no
Primitive yes
Minimal yes
Parity odd

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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([143,192]))
 
pari: [g,chi] = znchar(Mod(42,3025))
 

Basic properties

Modulus: \(3025\)
Conductor: \(3025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(220\)
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 3025.df

\(\chi_{3025}(42,\cdot)\) \(\chi_{3025}(48,\cdot)\) \(\chi_{3025}(108,\cdot)\) \(\chi_{3025}(113,\cdot)\) \(\chi_{3025}(203,\cdot)\) \(\chi_{3025}(212,\cdot)\) \(\chi_{3025}(247,\cdot)\) \(\chi_{3025}(317,\cdot)\) \(\chi_{3025}(383,\cdot)\) \(\chi_{3025}(388,\cdot)\) \(\chi_{3025}(477,\cdot)\) \(\chi_{3025}(478,\cdot)\) \(\chi_{3025}(522,\cdot)\) \(\chi_{3025}(592,\cdot)\) \(\chi_{3025}(598,\cdot)\) \(\chi_{3025}(658,\cdot)\) \(\chi_{3025}(663,\cdot)\) \(\chi_{3025}(752,\cdot)\) \(\chi_{3025}(762,\cdot)\) \(\chi_{3025}(797,\cdot)\) \(\chi_{3025}(867,\cdot)\) \(\chi_{3025}(873,\cdot)\) \(\chi_{3025}(933,\cdot)\) \(\chi_{3025}(938,\cdot)\) \(\chi_{3025}(1027,\cdot)\) \(\chi_{3025}(1028,\cdot)\) \(\chi_{3025}(1037,\cdot)\) \(\chi_{3025}(1072,\cdot)\) \(\chi_{3025}(1142,\cdot)\) \(\chi_{3025}(1148,\cdot)\) ...

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

\((727,2301)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{48}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 3025 }(42, a) \) \(-1\)\(1\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{48}{55}\right)\)\(e\left(\frac{79}{220}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{87}{220}\right)\)\(e\left(\frac{109}{220}\right)\)\(e\left(\frac{97}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3025 }(42,a) \;\) at \(\;a = \) e.g. 2