Properties

Label 980.711
Modulus $980$
Conductor $196$
Order $42$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,0,16]))
 
pari: [g,chi] = znchar(Mod(711,980))
 

Basic properties

Modulus: \(980\)
Conductor: \(196\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{196}(123,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 980.bm

\(\chi_{980}(11,\cdot)\) \(\chi_{980}(51,\cdot)\) \(\chi_{980}(151,\cdot)\) \(\chi_{980}(191,\cdot)\) \(\chi_{980}(291,\cdot)\) \(\chi_{980}(331,\cdot)\) \(\chi_{980}(431,\cdot)\) \(\chi_{980}(571,\cdot)\) \(\chi_{980}(611,\cdot)\) \(\chi_{980}(711,\cdot)\) \(\chi_{980}(751,\cdot)\) \(\chi_{980}(891,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.0.74252462132603256348231837398371002884673933378885582779211491265789772693504.1

Values on generators

\((491,197,101)\) → \((-1,1,e\left(\frac{8}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 980 }(711, a) \) \(-1\)\(1\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 980 }(711,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 980 }(711,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 980 }(711,·),\chi_{ 980 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 980 }(711,·)) \;\) at \(\; a,b = \) e.g. 1,2