Properties

Label 2025.16
Modulus $2025$
Conductor $2025$
Order $135$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2025, base_ring=CyclotomicField(270))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,54]))
 
pari: [g,chi] = znchar(Mod(16,2025))
 

Basic properties

Modulus: \(2025\)
Conductor: \(2025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(135\)
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 2025.bo

\(\chi_{2025}(16,\cdot)\) \(\chi_{2025}(31,\cdot)\) \(\chi_{2025}(61,\cdot)\) \(\chi_{2025}(106,\cdot)\) \(\chi_{2025}(121,\cdot)\) \(\chi_{2025}(166,\cdot)\) \(\chi_{2025}(196,\cdot)\) \(\chi_{2025}(211,\cdot)\) \(\chi_{2025}(241,\cdot)\) \(\chi_{2025}(256,\cdot)\) \(\chi_{2025}(286,\cdot)\) \(\chi_{2025}(331,\cdot)\) \(\chi_{2025}(346,\cdot)\) \(\chi_{2025}(391,\cdot)\) \(\chi_{2025}(421,\cdot)\) \(\chi_{2025}(436,\cdot)\) \(\chi_{2025}(466,\cdot)\) \(\chi_{2025}(481,\cdot)\) \(\chi_{2025}(511,\cdot)\) \(\chi_{2025}(556,\cdot)\) \(\chi_{2025}(571,\cdot)\) \(\chi_{2025}(616,\cdot)\) \(\chi_{2025}(646,\cdot)\) \(\chi_{2025}(661,\cdot)\) \(\chi_{2025}(691,\cdot)\) \(\chi_{2025}(706,\cdot)\) \(\chi_{2025}(736,\cdot)\) \(\chi_{2025}(781,\cdot)\) \(\chi_{2025}(796,\cdot)\) \(\chi_{2025}(841,\cdot)\) ...

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: $\Q(\zeta_{135})$
Fixed field: Number field defined by a degree 135 polynomial (not computed)

Values on generators

\((326,1702)\) → \((e\left(\frac{2}{27}\right),e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 2025 }(16, a) \) \(1\)\(1\)\(e\left(\frac{37}{135}\right)\)\(e\left(\frac{74}{135}\right)\)\(e\left(\frac{5}{27}\right)\)\(e\left(\frac{37}{45}\right)\)\(e\left(\frac{22}{135}\right)\)\(e\left(\frac{53}{135}\right)\)\(e\left(\frac{62}{135}\right)\)\(e\left(\frac{13}{135}\right)\)\(e\left(\frac{2}{45}\right)\)\(e\left(\frac{7}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2025 }(16,a) \;\) at \(\;a = \) e.g. 2