Properties

Label 8020.13
Modulus $8020$
Conductor $2005$
Order $400$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8020, base_ring=CyclotomicField(400))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,300,319]))
 
pari: [g,chi] = znchar(Mod(13,8020))
 

Basic properties

Modulus: \(8020\)
Conductor: \(2005\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(400\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2005}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8020.dz

\(\chi_{8020}(13,\cdot)\) \(\chi_{8020}(37,\cdot)\) \(\chi_{8020}(117,\cdot)\) \(\chi_{8020}(137,\cdot)\) \(\chi_{8020}(193,\cdot)\) \(\chi_{8020}(217,\cdot)\) \(\chi_{8020}(297,\cdot)\) \(\chi_{8020}(453,\cdot)\) \(\chi_{8020}(493,\cdot)\) \(\chi_{8020}(533,\cdot)\) \(\chi_{8020}(537,\cdot)\) \(\chi_{8020}(617,\cdot)\) \(\chi_{8020}(673,\cdot)\) \(\chi_{8020}(817,\cdot)\) \(\chi_{8020}(833,\cdot)\) \(\chi_{8020}(877,\cdot)\) \(\chi_{8020}(1053,\cdot)\) \(\chi_{8020}(1137,\cdot)\) \(\chi_{8020}(1157,\cdot)\) \(\chi_{8020}(1237,\cdot)\) \(\chi_{8020}(1257,\cdot)\) \(\chi_{8020}(1353,\cdot)\) \(\chi_{8020}(1397,\cdot)\) \(\chi_{8020}(1417,\cdot)\) \(\chi_{8020}(1517,\cdot)\) \(\chi_{8020}(1537,\cdot)\) \(\chi_{8020}(1573,\cdot)\) \(\chi_{8020}(1657,\cdot)\) \(\chi_{8020}(1733,\cdot)\) \(\chi_{8020}(1837,\cdot)\) ...

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: $\Q(\zeta_{400})$
Fixed field: Number field defined by a degree 400 polynomial (not computed)

Values on generators

\((4011,6417,7221)\) → \((1,-i,e\left(\frac{319}{400}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 8020 }(13, a) \) \(1\)\(1\)\(e\left(\frac{19}{400}\right)\)\(e\left(\frac{39}{200}\right)\)\(e\left(\frac{19}{200}\right)\)\(e\left(\frac{143}{200}\right)\)\(e\left(\frac{261}{400}\right)\)\(e\left(\frac{77}{400}\right)\)\(e\left(\frac{197}{400}\right)\)\(e\left(\frac{97}{400}\right)\)\(e\left(\frac{129}{400}\right)\)\(e\left(\frac{57}{400}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8020 }(13,a) \;\) at \(\;a = \) e.g. 2