from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(231, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,20,3]))
pari: [g,chi] = znchar(Mod(200,231))
Basic properties
| Modulus: | \(231\) | |
| Conductor: | \(231\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | 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 231.be
\(\chi_{231}(2,\cdot)\) \(\chi_{231}(74,\cdot)\) \(\chi_{231}(95,\cdot)\) \(\chi_{231}(107,\cdot)\) \(\chi_{231}(116,\cdot)\) \(\chi_{231}(128,\cdot)\) \(\chi_{231}(149,\cdot)\) \(\chi_{231}(200,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((155,199,211)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{1}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 231 }(200, a) \) | \(1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) |
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)