from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(220, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,5,12]))
pari: [g,chi] = znchar(Mod(207,220))
Basic properties
Modulus: | \(220\) | |
Conductor: | \(220\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | 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 220.v
\(\chi_{220}(3,\cdot)\) \(\chi_{220}(27,\cdot)\) \(\chi_{220}(47,\cdot)\) \(\chi_{220}(103,\cdot)\) \(\chi_{220}(147,\cdot)\) \(\chi_{220}(163,\cdot)\) \(\chi_{220}(203,\cdot)\) \(\chi_{220}(207,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((111,177,101)\) → \((-1,i,e\left(\frac{3}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 220 }(207, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) | \(i\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)