sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(529, base_ring=CyclotomicField(46))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([38]))
pari: [g,chi] = znchar(Mod(300,529))
Basic properties
| Modulus: | \(529\) | |
| Conductor: | \(529\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(23\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 529.e
\(\chi_{529}(24,\cdot)\) \(\chi_{529}(47,\cdot)\) \(\chi_{529}(70,\cdot)\) \(\chi_{529}(93,\cdot)\) \(\chi_{529}(116,\cdot)\) \(\chi_{529}(139,\cdot)\) \(\chi_{529}(162,\cdot)\) \(\chi_{529}(185,\cdot)\) \(\chi_{529}(208,\cdot)\) \(\chi_{529}(231,\cdot)\) \(\chi_{529}(254,\cdot)\) \(\chi_{529}(277,\cdot)\) \(\chi_{529}(300,\cdot)\) \(\chi_{529}(323,\cdot)\) \(\chi_{529}(346,\cdot)\) \(\chi_{529}(369,\cdot)\) \(\chi_{529}(392,\cdot)\) \(\chi_{529}(415,\cdot)\) \(\chi_{529}(438,\cdot)\) \(\chi_{529}(461,\cdot)\) \(\chi_{529}(484,\cdot)\) \(\chi_{529}(507,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
| Fixed field: | 23.23.824185149135487077883465900577270766354751717380230010246241.1 |
Values on generators
\(5\) → \(e\left(\frac{19}{23}\right)\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \(1\) | \(1\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{529}(300,\cdot)) = \sum_{r\in \Z/529\Z} \chi_{529}(300,r) e\left(\frac{2r}{529}\right) = 22.9854003383+0.8193724951i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{529}(300,\cdot),\chi_{529}(1,\cdot)) = \sum_{r\in \Z/529\Z} \chi_{529}(300,r) \chi_{529}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{529}(300,·))
= \sum_{r \in \Z/529\Z}
\chi_{529}(300,r) e\left(\frac{1 r + 2 r^{-1}}{529}\right)
= 0.0 \)