Properties

Label 3025.23
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([121,140]))
 
pari: [g,chi] = znchar(Mod(23,3025))
 

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.dc

\(\chi_{3025}(12,\cdot)\) \(\chi_{3025}(23,\cdot)\) \(\chi_{3025}(67,\cdot)\) \(\chi_{3025}(78,\cdot)\) \(\chi_{3025}(133,\cdot)\) \(\chi_{3025}(177,\cdot)\) \(\chi_{3025}(188,\cdot)\) \(\chi_{3025}(287,\cdot)\) \(\chi_{3025}(298,\cdot)\) \(\chi_{3025}(342,\cdot)\) \(\chi_{3025}(353,\cdot)\) \(\chi_{3025}(397,\cdot)\) \(\chi_{3025}(408,\cdot)\) \(\chi_{3025}(452,\cdot)\) \(\chi_{3025}(463,\cdot)\) \(\chi_{3025}(562,\cdot)\) \(\chi_{3025}(573,\cdot)\) \(\chi_{3025}(617,\cdot)\) \(\chi_{3025}(628,\cdot)\) \(\chi_{3025}(672,\cdot)\) \(\chi_{3025}(683,\cdot)\) \(\chi_{3025}(738,\cdot)\) \(\chi_{3025}(837,\cdot)\) \(\chi_{3025}(892,\cdot)\) \(\chi_{3025}(903,\cdot)\) \(\chi_{3025}(947,\cdot)\) \(\chi_{3025}(958,\cdot)\) \(\chi_{3025}(1002,\cdot)\) \(\chi_{3025}(1013,\cdot)\) \(\chi_{3025}(1112,\cdot)\) ...

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

\((727,2301)\) → \((e\left(\frac{11}{20}\right),e\left(\frac{7}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 3025 }(23, a) \) \(-1\)\(1\)\(e\left(\frac{41}{220}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{41}{110}\right)\)\(e\left(\frac{2}{55}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{123}{220}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{49}{220}\right)\)\(e\left(\frac{159}{220}\right)\)\(e\left(\frac{43}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3025 }(23,a) \;\) at \(\;a = \) e.g. 2