Properties

Label 4025.1111
Modulus $4025$
Conductor $4025$
Order $330$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4025, base_ring=CyclotomicField(330))
 
M = H._module
 
chi = DirichletCharacter(H, M([264,275,285]))
 
pari: [g,chi] = znchar(Mod(1111,4025))
 

Basic properties

Modulus: \(4025\)
Conductor: \(4025\)
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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4025.dj

\(\chi_{4025}(61,\cdot)\) \(\chi_{4025}(66,\cdot)\) \(\chi_{4025}(136,\cdot)\) \(\chi_{4025}(166,\cdot)\) \(\chi_{4025}(171,\cdot)\) \(\chi_{4025}(241,\cdot)\) \(\chi_{4025}(306,\cdot)\) \(\chi_{4025}(341,\cdot)\) \(\chi_{4025}(411,\cdot)\) \(\chi_{4025}(481,\cdot)\) \(\chi_{4025}(516,\cdot)\) \(\chi_{4025}(521,\cdot)\) \(\chi_{4025}(586,\cdot)\) \(\chi_{4025}(661,\cdot)\) \(\chi_{4025}(766,\cdot)\) \(\chi_{4025}(796,\cdot)\) \(\chi_{4025}(866,\cdot)\) \(\chi_{4025}(871,\cdot)\) \(\chi_{4025}(941,\cdot)\) \(\chi_{4025}(971,\cdot)\) \(\chi_{4025}(1006,\cdot)\) \(\chi_{4025}(1046,\cdot)\) \(\chi_{4025}(1111,\cdot)\) \(\chi_{4025}(1146,\cdot)\) \(\chi_{4025}(1216,\cdot)\) \(\chi_{4025}(1256,\cdot)\) \(\chi_{4025}(1286,\cdot)\) \(\chi_{4025}(1321,\cdot)\) \(\chi_{4025}(1391,\cdot)\) \(\chi_{4025}(1431,\cdot)\) ...

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

\((2577,1151,3501)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{5}{6}\right),e\left(\frac{19}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(1111, a) \) \(1\)\(1\)\(e\left(\frac{32}{165}\right)\)\(e\left(\frac{83}{330}\right)\)\(e\left(\frac{64}{165}\right)\)\(e\left(\frac{49}{110}\right)\)\(e\left(\frac{32}{55}\right)\)\(e\left(\frac{83}{165}\right)\)\(e\left(\frac{299}{330}\right)\)\(e\left(\frac{211}{330}\right)\)\(e\left(\frac{87}{110}\right)\)\(e\left(\frac{128}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(1111,a) \;\) at \(\;a = \) e.g. 2