sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(75, base_ring=CyclotomicField(20))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,17]))
pari: [g,chi] = znchar(Mod(22,75))
Basic properties
| Modulus: | \(75\) | |
| Conductor: | \(25\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{25}(22,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 75.k
\(\chi_{75}(13,\cdot)\) \(\chi_{75}(22,\cdot)\) \(\chi_{75}(28,\cdot)\) \(\chi_{75}(37,\cdot)\) \(\chi_{75}(52,\cdot)\) \(\chi_{75}(58,\cdot)\) \(\chi_{75}(67,\cdot)\) \(\chi_{75}(73,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | \(\Q(\zeta_{25})\) |
Values on generators
\((26,52)\) → \((1,e\left(\frac{17}{20}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \(-1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) |
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)