Properties

Label 4000.311
Modulus $4000$
Conductor $2000$
Order $100$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(100))
 
M = H._module
 
chi = DirichletCharacter(H, M([50,75,56]))
 
pari: [g,chi] = znchar(Mod(311,4000))
 

Basic properties

Modulus: \(4000\)
Conductor: \(2000\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2000}(1811,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4000.cm

\(\chi_{4000}(71,\cdot)\) \(\chi_{4000}(231,\cdot)\) \(\chi_{4000}(311,\cdot)\) \(\chi_{4000}(391,\cdot)\) \(\chi_{4000}(471,\cdot)\) \(\chi_{4000}(631,\cdot)\) \(\chi_{4000}(711,\cdot)\) \(\chi_{4000}(791,\cdot)\) \(\chi_{4000}(871,\cdot)\) \(\chi_{4000}(1031,\cdot)\) \(\chi_{4000}(1111,\cdot)\) \(\chi_{4000}(1191,\cdot)\) \(\chi_{4000}(1271,\cdot)\) \(\chi_{4000}(1431,\cdot)\) \(\chi_{4000}(1511,\cdot)\) \(\chi_{4000}(1591,\cdot)\) \(\chi_{4000}(1671,\cdot)\) \(\chi_{4000}(1831,\cdot)\) \(\chi_{4000}(1911,\cdot)\) \(\chi_{4000}(1991,\cdot)\) \(\chi_{4000}(2071,\cdot)\) \(\chi_{4000}(2231,\cdot)\) \(\chi_{4000}(2311,\cdot)\) \(\chi_{4000}(2391,\cdot)\) \(\chi_{4000}(2471,\cdot)\) \(\chi_{4000}(2631,\cdot)\) \(\chi_{4000}(2711,\cdot)\) \(\chi_{4000}(2791,\cdot)\) \(\chi_{4000}(2871,\cdot)\) \(\chi_{4000}(3031,\cdot)\) ...

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: $\Q(\zeta_{100})$
Fixed field: Number field defined by a degree 100 polynomial

Values on generators

\((2751,2501,1377)\) → \((-1,-i,e\left(\frac{14}{25}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(311, a) \) \(-1\)\(1\)\(e\left(\frac{67}{100}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{17}{50}\right)\)\(e\left(\frac{81}{100}\right)\)\(e\left(\frac{9}{100}\right)\)\(e\left(\frac{22}{25}\right)\)\(e\left(\frac{83}{100}\right)\)\(e\left(\frac{27}{100}\right)\)\(e\left(\frac{9}{25}\right)\)\(e\left(\frac{1}{100}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(311,a) \;\) at \(\;a = \) e.g. 2