Properties

Label 8020.307
Modulus $8020$
Conductor $8020$
Order $100$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8020, base_ring=CyclotomicField(100))
 
M = H._module
 
chi = DirichletCharacter(H, M([50,25,11]))
 
pari: [g,chi] = znchar(Mod(307,8020))
 

Basic properties

Modulus: \(8020\)
Conductor: \(8020\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8020.dg

\(\chi_{8020}(307,\cdot)\) \(\chi_{8020}(383,\cdot)\) \(\chi_{8020}(883,\cdot)\) \(\chi_{8020}(923,\cdot)\) \(\chi_{8020}(1047,\cdot)\) \(\chi_{8020}(1267,\cdot)\) \(\chi_{8020}(1303,\cdot)\) \(\chi_{8020}(1547,\cdot)\) \(\chi_{8020}(1703,\cdot)\) \(\chi_{8020}(1787,\cdot)\) \(\chi_{8020}(2223,\cdot)\) \(\chi_{8020}(2307,\cdot)\) \(\chi_{8020}(2463,\cdot)\) \(\chi_{8020}(2707,\cdot)\) \(\chi_{8020}(2743,\cdot)\) \(\chi_{8020}(2963,\cdot)\) \(\chi_{8020}(3087,\cdot)\) \(\chi_{8020}(3127,\cdot)\) \(\chi_{8020}(3627,\cdot)\) \(\chi_{8020}(3703,\cdot)\) \(\chi_{8020}(4207,\cdot)\) \(\chi_{8020}(4407,\cdot)\) \(\chi_{8020}(4527,\cdot)\) \(\chi_{8020}(4663,\cdot)\) \(\chi_{8020}(4763,\cdot)\) \(\chi_{8020}(5103,\cdot)\) \(\chi_{8020}(5123,\cdot)\) \(\chi_{8020}(5683,\cdot)\) \(\chi_{8020}(5707,\cdot)\) \(\chi_{8020}(5727,\cdot)\) ...

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

\((4011,6417,7221)\) → \((-1,i,e\left(\frac{11}{100}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 8020 }(307, a) \) \(1\)\(1\)\(e\left(\frac{9}{25}\right)\)\(e\left(\frac{57}{100}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{21}{25}\right)\)\(e\left(\frac{21}{25}\right)\)\(e\left(\frac{19}{50}\right)\)\(e\left(\frac{93}{100}\right)\)\(e\left(\frac{93}{100}\right)\)\(e\left(\frac{13}{50}\right)\)\(e\left(\frac{2}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8020 }(307,a) \;\) at \(\;a = \) e.g. 2