Properties

Label 9800.107
Modulus $9800$
Conductor $1960$
Order $84$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9800, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([42,42,21,4]))
 
pari: [g,chi] = znchar(Mod(107,9800))
 

Basic properties

Modulus: \(9800\)
Conductor: \(1960\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1960}(107,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9800.gc

\(\chi_{9800}(107,\cdot)\) \(\chi_{9800}(443,\cdot)\) \(\chi_{9800}(907,\cdot)\) \(\chi_{9800}(1507,\cdot)\) \(\chi_{9800}(2307,\cdot)\) \(\chi_{9800}(2643,\cdot)\) \(\chi_{9800}(2907,\cdot)\) \(\chi_{9800}(3243,\cdot)\) \(\chi_{9800}(3707,\cdot)\) \(\chi_{9800}(4043,\cdot)\) \(\chi_{9800}(4307,\cdot)\) \(\chi_{9800}(4643,\cdot)\) \(\chi_{9800}(5107,\cdot)\) \(\chi_{9800}(5443,\cdot)\) \(\chi_{9800}(5707,\cdot)\) \(\chi_{9800}(6043,\cdot)\) \(\chi_{9800}(6507,\cdot)\) \(\chi_{9800}(6843,\cdot)\) \(\chi_{9800}(7107,\cdot)\) \(\chi_{9800}(7443,\cdot)\) \(\chi_{9800}(8243,\cdot)\) \(\chi_{9800}(8843,\cdot)\) \(\chi_{9800}(9307,\cdot)\) \(\chi_{9800}(9643,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((7351,4901,1177,5001)\) → \((-1,-1,i,e\left(\frac{1}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 9800 }(107, a) \) \(1\)\(1\)\(e\left(\frac{67}{84}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{37}{84}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{84}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9800 }(107,a) \;\) at \(\;a = \) e.g. 2