Properties

Label 4025.1387
Modulus $4025$
Conductor $575$
Order $220$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4025, base_ring=CyclotomicField(220))
 
M = H._module
 
chi = DirichletCharacter(H, M([99,0,190]))
 
pari: [g,chi] = znchar(Mod(1387,4025))
 

Basic properties

Modulus: \(4025\)
Conductor: \(575\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(220\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{575}(237,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4025.df

\(\chi_{4025}(113,\cdot)\) \(\chi_{4025}(148,\cdot)\) \(\chi_{4025}(267,\cdot)\) \(\chi_{4025}(337,\cdot)\) \(\chi_{4025}(428,\cdot)\) \(\chi_{4025}(442,\cdot)\) \(\chi_{4025}(477,\cdot)\) \(\chi_{4025}(498,\cdot)\) \(\chi_{4025}(603,\cdot)\) \(\chi_{4025}(617,\cdot)\) \(\chi_{4025}(638,\cdot)\) \(\chi_{4025}(687,\cdot)\) \(\chi_{4025}(778,\cdot)\) \(\chi_{4025}(792,\cdot)\) \(\chi_{4025}(848,\cdot)\) \(\chi_{4025}(862,\cdot)\) \(\chi_{4025}(953,\cdot)\) \(\chi_{4025}(1023,\cdot)\) \(\chi_{4025}(1072,\cdot)\) \(\chi_{4025}(1142,\cdot)\) \(\chi_{4025}(1233,\cdot)\) \(\chi_{4025}(1247,\cdot)\) \(\chi_{4025}(1303,\cdot)\) \(\chi_{4025}(1387,\cdot)\) \(\chi_{4025}(1408,\cdot)\) \(\chi_{4025}(1422,\cdot)\) \(\chi_{4025}(1492,\cdot)\) \(\chi_{4025}(1548,\cdot)\) \(\chi_{4025}(1562,\cdot)\) \(\chi_{4025}(1583,\cdot)\) ...

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

\((2577,1151,3501)\) → \((e\left(\frac{9}{20}\right),1,e\left(\frac{19}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(1387, a) \) \(1\)\(1\)\(e\left(\frac{39}{220}\right)\)\(e\left(\frac{213}{220}\right)\)\(e\left(\frac{39}{110}\right)\)\(e\left(\frac{8}{55}\right)\)\(e\left(\frac{117}{220}\right)\)\(e\left(\frac{103}{110}\right)\)\(e\left(\frac{107}{110}\right)\)\(e\left(\frac{71}{220}\right)\)\(e\left(\frac{141}{220}\right)\)\(e\left(\frac{39}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(1387,a) \;\) at \(\;a = \) e.g. 2