Properties

Label 980.233
Modulus $980$
Conductor $245$
Order $84$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(980, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,63,64]))
 
pari: [g,chi] = znchar(Mod(233,980))
 

Basic properties

Modulus: \(980\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(233,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 980.bv

\(\chi_{980}(37,\cdot)\) \(\chi_{980}(53,\cdot)\) \(\chi_{980}(93,\cdot)\) \(\chi_{980}(137,\cdot)\) \(\chi_{980}(193,\cdot)\) \(\chi_{980}(233,\cdot)\) \(\chi_{980}(277,\cdot)\) \(\chi_{980}(317,\cdot)\) \(\chi_{980}(333,\cdot)\) \(\chi_{980}(417,\cdot)\) \(\chi_{980}(457,\cdot)\) \(\chi_{980}(473,\cdot)\) \(\chi_{980}(513,\cdot)\) \(\chi_{980}(597,\cdot)\) \(\chi_{980}(613,\cdot)\) \(\chi_{980}(653,\cdot)\) \(\chi_{980}(697,\cdot)\) \(\chi_{980}(737,\cdot)\) \(\chi_{980}(793,\cdot)\) \(\chi_{980}(837,\cdot)\) \(\chi_{980}(877,\cdot)\) \(\chi_{980}(893,\cdot)\) \(\chi_{980}(933,\cdot)\) \(\chi_{980}(977,\cdot)\)

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

\((491,197,101)\) → \((1,-i,e\left(\frac{16}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 980 }(233, a) \) \(-1\)\(1\)\(e\left(\frac{1}{84}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{67}{84}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{17}{84}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 980 }(233,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 980 }(233,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 980 }(233,·),\chi_{ 980 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 980 }(233,·)) \;\) at \(\; a,b = \) e.g. 1,2