Properties

Label 1944.7
Modulus $1944$
Conductor $972$
Order $162$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1944, base_ring=CyclotomicField(162))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([81,0,70]))
 
pari: [g,chi] = znchar(Mod(7,1944))
 

Basic properties

Modulus: \(1944\)
Conductor: \(972\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(162\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{972}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1944.bi

\(\chi_{1944}(7,\cdot)\) \(\chi_{1944}(31,\cdot)\) \(\chi_{1944}(79,\cdot)\) \(\chi_{1944}(103,\cdot)\) \(\chi_{1944}(151,\cdot)\) \(\chi_{1944}(175,\cdot)\) \(\chi_{1944}(223,\cdot)\) \(\chi_{1944}(247,\cdot)\) \(\chi_{1944}(295,\cdot)\) \(\chi_{1944}(319,\cdot)\) \(\chi_{1944}(367,\cdot)\) \(\chi_{1944}(391,\cdot)\) \(\chi_{1944}(439,\cdot)\) \(\chi_{1944}(463,\cdot)\) \(\chi_{1944}(511,\cdot)\) \(\chi_{1944}(535,\cdot)\) \(\chi_{1944}(583,\cdot)\) \(\chi_{1944}(607,\cdot)\) \(\chi_{1944}(655,\cdot)\) \(\chi_{1944}(679,\cdot)\) \(\chi_{1944}(727,\cdot)\) \(\chi_{1944}(751,\cdot)\) \(\chi_{1944}(799,\cdot)\) \(\chi_{1944}(823,\cdot)\) \(\chi_{1944}(871,\cdot)\) \(\chi_{1944}(895,\cdot)\) \(\chi_{1944}(943,\cdot)\) \(\chi_{1944}(967,\cdot)\) \(\chi_{1944}(1015,\cdot)\) \(\chi_{1944}(1039,\cdot)\) ...

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: $\Q(\zeta_{81})$
Fixed field: Number field defined by a degree 162 polynomial (not computed)

Values on generators

\((487,973,1217)\) → \((-1,1,e\left(\frac{35}{81}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{76}{81}\right)\)\(e\left(\frac{121}{162}\right)\)\(e\left(\frac{127}{162}\right)\)\(e\left(\frac{37}{81}\right)\)\(e\left(\frac{7}{27}\right)\)\(e\left(\frac{49}{54}\right)\)\(e\left(\frac{95}{162}\right)\)\(e\left(\frac{71}{81}\right)\)\(e\left(\frac{80}{81}\right)\)\(e\left(\frac{23}{162}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1944 }(7,a) \;\) at \(\;a = \) e.g. 2