from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(675, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,57]))
pari: [g,chi] = znchar(Mod(638,675))
Basic properties
| Modulus: | \(675\) | |
| Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{225}(113,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 675.bd
\(\chi_{675}(8,\cdot)\) \(\chi_{675}(17,\cdot)\) \(\chi_{675}(62,\cdot)\) \(\chi_{675}(98,\cdot)\) \(\chi_{675}(152,\cdot)\) \(\chi_{675}(197,\cdot)\) \(\chi_{675}(233,\cdot)\) \(\chi_{675}(278,\cdot)\) \(\chi_{675}(287,\cdot)\) \(\chi_{675}(413,\cdot)\) \(\chi_{675}(422,\cdot)\) \(\chi_{675}(467,\cdot)\) \(\chi_{675}(503,\cdot)\) \(\chi_{675}(548,\cdot)\) \(\chi_{675}(602,\cdot)\) \(\chi_{675}(638,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((326,352)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{19}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 675 }(638, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
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)