Properties

Label 4000.1273
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([0,25,31]))
 
pari: [g,chi] = znchar(Mod(1273,4000))
 

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 \(\chi_{2000}(773,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4000.cu

\(\chi_{4000}(137,\cdot)\) \(\chi_{4000}(153,\cdot)\) \(\chi_{4000}(297,\cdot)\) \(\chi_{4000}(313,\cdot)\) \(\chi_{4000}(473,\cdot)\) \(\chi_{4000}(617,\cdot)\) \(\chi_{4000}(633,\cdot)\) \(\chi_{4000}(777,\cdot)\) \(\chi_{4000}(937,\cdot)\) \(\chi_{4000}(953,\cdot)\) \(\chi_{4000}(1097,\cdot)\) \(\chi_{4000}(1113,\cdot)\) \(\chi_{4000}(1273,\cdot)\) \(\chi_{4000}(1417,\cdot)\) \(\chi_{4000}(1433,\cdot)\) \(\chi_{4000}(1577,\cdot)\) \(\chi_{4000}(1737,\cdot)\) \(\chi_{4000}(1753,\cdot)\) \(\chi_{4000}(1897,\cdot)\) \(\chi_{4000}(1913,\cdot)\) \(\chi_{4000}(2073,\cdot)\) \(\chi_{4000}(2217,\cdot)\) \(\chi_{4000}(2233,\cdot)\) \(\chi_{4000}(2377,\cdot)\) \(\chi_{4000}(2537,\cdot)\) \(\chi_{4000}(2553,\cdot)\) \(\chi_{4000}(2697,\cdot)\) \(\chi_{4000}(2713,\cdot)\) \(\chi_{4000}(2873,\cdot)\) \(\chi_{4000}(3017,\cdot)\) ...

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

\((2751,2501,1377)\) → \((1,i,e\left(\frac{31}{100}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4000 }(1273, a) \) \(-1\)\(1\)\(e\left(\frac{23}{25}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{21}{25}\right)\)\(e\left(\frac{81}{100}\right)\)\(e\left(\frac{21}{25}\right)\)\(e\left(\frac{63}{100}\right)\)\(e\left(\frac{33}{100}\right)\)\(e\left(\frac{77}{100}\right)\)\(e\left(\frac{11}{100}\right)\)\(e\left(\frac{19}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4000 }(1273,a) \;\) at \(\;a = \) e.g. 2