sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(99, base_ring=CyclotomicField(30))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([20,27]))
pari: [g,chi] = znchar(Mod(61,99))
Basic properties
| Modulus: | \(99\) | |
| Conductor: | \(99\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 99.o
\(\chi_{99}(7,\cdot)\) \(\chi_{99}(13,\cdot)\) \(\chi_{99}(40,\cdot)\) \(\chi_{99}(52,\cdot)\) \(\chi_{99}(61,\cdot)\) \(\chi_{99}(79,\cdot)\) \(\chi_{99}(85,\cdot)\) \(\chi_{99}(94,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.0.159386923550435671074967363509984324121230045171.1 |
Values on generators
\((56,46)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{9}{10}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \(-1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{99}(61,\cdot)) = \sum_{r\in \Z/99\Z} \chi_{99}(61,r) e\left(\frac{2r}{99}\right) = -8.3200196391+-5.4568556151i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{99}(61,\cdot),\chi_{99}(1,\cdot)) = \sum_{r\in \Z/99\Z} \chi_{99}(61,r) \chi_{99}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{99}(61,·))
= \sum_{r \in \Z/99\Z}
\chi_{99}(61,r) e\left(\frac{1 r + 2 r^{-1}}{99}\right)
= -6.4928926992+-1.3801069474i \)