Properties

Label 980.967
Modulus $980$
Conductor $980$
Order $28$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(980, base_ring=CyclotomicField(28))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([14,7,8]))
 
pari: [g,chi] = znchar(Mod(967,980))
 

Basic properties

Modulus: \(980\)
Conductor: \(980\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
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 980.bk

\(\chi_{980}(43,\cdot)\) \(\chi_{980}(127,\cdot)\) \(\chi_{980}(183,\cdot)\) \(\chi_{980}(267,\cdot)\) \(\chi_{980}(323,\cdot)\) \(\chi_{980}(407,\cdot)\) \(\chi_{980}(463,\cdot)\) \(\chi_{980}(547,\cdot)\) \(\chi_{980}(603,\cdot)\) \(\chi_{980}(743,\cdot)\) \(\chi_{980}(827,\cdot)\) \(\chi_{980}(967,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{28})\)
Fixed field: 28.28.4698031131813648056477467012100308504166528000000000000000000000.1

Values on generators

\((491,197,101)\) → \((-1,i,e\left(\frac{2}{7}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{11}{28}\right)\)\(1\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{9}{14}\right)\)\(-1\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 980 }(967,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{980}(967,\cdot)) = \sum_{r\in \Z/980\Z} \chi_{980}(967,r) e\left(\frac{r}{490}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 980 }(967,·),\chi_{ 980 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{980}(967,\cdot),\chi_{980}(1,\cdot)) = \sum_{r\in \Z/980\Z} \chi_{980}(967,r) \chi_{980}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 980 }(967,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{980}(967,·)) = \sum_{r \in \Z/980\Z} \chi_{980}(967,r) e\left(\frac{1 r + 2 r^{-1}}{980}\right) = 0.0 \)