from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(572, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,18,5]))
pari: [g,chi] = znchar(Mod(41,572))
Basic properties
| Modulus: | \(572\) | |
| Conductor: | \(143\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{143}(41,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 572.bv
\(\chi_{572}(41,\cdot)\) \(\chi_{572}(85,\cdot)\) \(\chi_{572}(145,\cdot)\) \(\chi_{572}(149,\cdot)\) \(\chi_{572}(189,\cdot)\) \(\chi_{572}(193,\cdot)\) \(\chi_{572}(249,\cdot)\) \(\chi_{572}(293,\cdot)\) \(\chi_{572}(305,\cdot)\) \(\chi_{572}(349,\cdot)\) \(\chi_{572}(409,\cdot)\) \(\chi_{572}(453,\cdot)\) \(\chi_{572}(457,\cdot)\) \(\chi_{572}(501,\cdot)\) \(\chi_{572}(513,\cdot)\) \(\chi_{572}(557,\cdot)\)
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
\((287,365,353)\) → \((1,e\left(\frac{3}{10}\right),e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 572 }(41, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(-i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{9}{10}\right)\) |
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)