from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(987, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,0,8]))
pari: [g,chi] = znchar(Mod(8,987))
Basic properties
| Modulus: | \(987\) | |
| Conductor: | \(141\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(46\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{141}(8,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 987.x
\(\chi_{987}(8,\cdot)\) \(\chi_{987}(50,\cdot)\) \(\chi_{987}(71,\cdot)\) \(\chi_{987}(155,\cdot)\) \(\chi_{987}(197,\cdot)\) \(\chi_{987}(239,\cdot)\) \(\chi_{987}(260,\cdot)\) \(\chi_{987}(365,\cdot)\) \(\chi_{987}(491,\cdot)\) \(\chi_{987}(512,\cdot)\) \(\chi_{987}(533,\cdot)\) \(\chi_{987}(554,\cdot)\) \(\chi_{987}(596,\cdot)\) \(\chi_{987}(617,\cdot)\) \(\chi_{987}(638,\cdot)\) \(\chi_{987}(722,\cdot)\) \(\chi_{987}(764,\cdot)\) \(\chi_{987}(806,\cdot)\) \(\chi_{987}(827,\cdot)\) \(\chi_{987}(848,\cdot)\) \(\chi_{987}(911,\cdot)\) \(\chi_{987}(974,\cdot)\)
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: | 46.0.3516370336176915030779886015601767871077707157889593350075735586626118367196692091787.1 |
Values on generators
\((659,283,757)\) → \((-1,1,e\left(\frac{4}{23}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 987 }(8, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{19}{23}\right)\) |
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)