from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(987, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([0,0,24]))
pari: [g,chi] = znchar(Mod(841,987))
Basic properties
| Modulus: | \(987\) | |
| Conductor: | \(47\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(23\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{47}(42,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 987.q
\(\chi_{987}(64,\cdot)\) \(\chi_{987}(106,\cdot)\) \(\chi_{987}(148,\cdot)\) \(\chi_{987}(169,\cdot)\) \(\chi_{987}(190,\cdot)\) \(\chi_{987}(253,\cdot)\) \(\chi_{987}(316,\cdot)\) \(\chi_{987}(337,\cdot)\) \(\chi_{987}(379,\cdot)\) \(\chi_{987}(400,\cdot)\) \(\chi_{987}(484,\cdot)\) \(\chi_{987}(526,\cdot)\) \(\chi_{987}(568,\cdot)\) \(\chi_{987}(589,\cdot)\) \(\chi_{987}(694,\cdot)\) \(\chi_{987}(820,\cdot)\) \(\chi_{987}(841,\cdot)\) \(\chi_{987}(862,\cdot)\) \(\chi_{987}(883,\cdot)\) \(\chi_{987}(925,\cdot)\) \(\chi_{987}(946,\cdot)\) \(\chi_{987}(967,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
| Fixed field: | Number field defined by a degree 23 polynomial |
Values on generators
\((659,283,757)\) → \((1,1,e\left(\frac{12}{23}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 987 }(841, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) |
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)