Properties

Label 980.937
Modulus $980$
Conductor $245$
Order $28$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: 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([0,7,18]))
 
pari: [g,chi] = znchar(Mod(937,980))
 

Basic properties

Modulus: \(980\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(202,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 980.bi

\(\chi_{980}(13,\cdot)\) \(\chi_{980}(153,\cdot)\) \(\chi_{980}(237,\cdot)\) \(\chi_{980}(377,\cdot)\) \(\chi_{980}(433,\cdot)\) \(\chi_{980}(517,\cdot)\) \(\chi_{980}(573,\cdot)\) \(\chi_{980}(657,\cdot)\) \(\chi_{980}(713,\cdot)\) \(\chi_{980}(797,\cdot)\) \(\chi_{980}(853,\cdot)\) \(\chi_{980}(937,\cdot)\)

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: 28.28.857574960063654015836849375042748140931725025177001953125.1

Values on generators

\((491,197,101)\) → \((1,i,e\left(\frac{9}{14}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 980 }(937, a) \) \(1\)\(1\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{9}{28}\right)\)\(1\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{1}{14}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 980 }(937,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 980 }(937,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 980 }(937,·),\chi_{ 980 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 980 }(937,·)) \;\) at \(\; a,b = \) e.g. 1,2