Properties

Label 4025.41
Modulus $4025$
Conductor $4025$
Order $110$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4025, base_ring=CyclotomicField(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,55,60]))
 
pari: [g,chi] = znchar(Mod(41,4025))
 

Basic properties

Modulus: \(4025\)
Conductor: \(4025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
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 4025.cw

\(\chi_{4025}(6,\cdot)\) \(\chi_{4025}(41,\cdot)\) \(\chi_{4025}(146,\cdot)\) \(\chi_{4025}(216,\cdot)\) \(\chi_{4025}(496,\cdot)\) \(\chi_{4025}(531,\cdot)\) \(\chi_{4025}(671,\cdot)\) \(\chi_{4025}(706,\cdot)\) \(\chi_{4025}(811,\cdot)\) \(\chi_{4025}(846,\cdot)\) \(\chi_{4025}(1021,\cdot)\) \(\chi_{4025}(1231,\cdot)\) \(\chi_{4025}(1336,\cdot)\) \(\chi_{4025}(1406,\cdot)\) \(\chi_{4025}(1511,\cdot)\) \(\chi_{4025}(1616,\cdot)\) \(\chi_{4025}(1756,\cdot)\) \(\chi_{4025}(2036,\cdot)\) \(\chi_{4025}(2106,\cdot)\) \(\chi_{4025}(2141,\cdot)\) \(\chi_{4025}(2211,\cdot)\) \(\chi_{4025}(2281,\cdot)\) \(\chi_{4025}(2316,\cdot)\) \(\chi_{4025}(2421,\cdot)\) \(\chi_{4025}(2456,\cdot)\) \(\chi_{4025}(2561,\cdot)\) \(\chi_{4025}(2631,\cdot)\) \(\chi_{4025}(2841,\cdot)\) \(\chi_{4025}(2911,\cdot)\) \(\chi_{4025}(2946,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((2577,1151,3501)\) → \((e\left(\frac{1}{5}\right),-1,e\left(\frac{6}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(41, a) \) \(-1\)\(1\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{69}{110}\right)\)\(e\left(\frac{32}{55}\right)\)\(e\left(\frac{101}{110}\right)\)\(e\left(\frac{48}{55}\right)\)\(e\left(\frac{14}{55}\right)\)\(e\left(\frac{6}{55}\right)\)\(e\left(\frac{23}{110}\right)\)\(e\left(\frac{103}{110}\right)\)\(e\left(\frac{9}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(41,a) \;\) at \(\;a = \) e.g. 2