Properties

Label 4025.652
Modulus $4025$
Conductor $575$
Order $220$
Real no
Primitive no
Minimal yes
Parity odd

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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([11,0,60]))
 
pari: [g,chi] = znchar(Mod(652,4025))
 

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 \(\chi_{575}(77,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4025.dg

\(\chi_{4025}(8,\cdot)\) \(\chi_{4025}(78,\cdot)\) \(\chi_{4025}(127,\cdot)\) \(\chi_{4025}(197,\cdot)\) \(\chi_{4025}(288,\cdot)\) \(\chi_{4025}(302,\cdot)\) \(\chi_{4025}(358,\cdot)\) \(\chi_{4025}(372,\cdot)\) \(\chi_{4025}(463,\cdot)\) \(\chi_{4025}(512,\cdot)\) \(\chi_{4025}(533,\cdot)\) \(\chi_{4025}(547,\cdot)\) \(\chi_{4025}(652,\cdot)\) \(\chi_{4025}(673,\cdot)\) \(\chi_{4025}(708,\cdot)\) \(\chi_{4025}(722,\cdot)\) \(\chi_{4025}(813,\cdot)\) \(\chi_{4025}(883,\cdot)\) \(\chi_{4025}(1002,\cdot)\) \(\chi_{4025}(1037,\cdot)\) \(\chi_{4025}(1163,\cdot)\) \(\chi_{4025}(1177,\cdot)\) \(\chi_{4025}(1198,\cdot)\) \(\chi_{4025}(1212,\cdot)\) \(\chi_{4025}(1317,\cdot)\) \(\chi_{4025}(1338,\cdot)\) \(\chi_{4025}(1352,\cdot)\) \(\chi_{4025}(1373,\cdot)\) \(\chi_{4025}(1478,\cdot)\) \(\chi_{4025}(1513,\cdot)\) ...

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

\((2577,1151,3501)\) → \((e\left(\frac{1}{20}\right),1,e\left(\frac{3}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 4025 }(652, a) \) \(-1\)\(1\)\(e\left(\frac{131}{220}\right)\)\(e\left(\frac{157}{220}\right)\)\(e\left(\frac{21}{110}\right)\)\(e\left(\frac{17}{55}\right)\)\(e\left(\frac{173}{220}\right)\)\(e\left(\frac{47}{110}\right)\)\(e\left(\frac{14}{55}\right)\)\(e\left(\frac{199}{220}\right)\)\(e\left(\frac{169}{220}\right)\)\(e\left(\frac{21}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4025 }(652,a) \;\) at \(\;a = \) e.g. 2