sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(175, base_ring=CyclotomicField(60))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([27,20]))
pari: [g,chi] = znchar(Mod(37,175))
Basic properties
| Modulus: | \(175\) | |
| Conductor: | \(175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 175.w
\(\chi_{175}(2,\cdot)\) \(\chi_{175}(23,\cdot)\) \(\chi_{175}(37,\cdot)\) \(\chi_{175}(53,\cdot)\) \(\chi_{175}(58,\cdot)\) \(\chi_{175}(67,\cdot)\) \(\chi_{175}(72,\cdot)\) \(\chi_{175}(88,\cdot)\) \(\chi_{175}(102,\cdot)\) \(\chi_{175}(123,\cdot)\) \(\chi_{175}(128,\cdot)\) \(\chi_{175}(137,\cdot)\) \(\chi_{175}(142,\cdot)\) \(\chi_{175}(158,\cdot)\) \(\chi_{175}(163,\cdot)\) \(\chi_{175}(172,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((127,101)\) → \((e\left(\frac{9}{20}\right),e\left(\frac{1}{3}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \(-1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{175}(37,\cdot)) = \sum_{r\in \Z/175\Z} \chi_{175}(37,r) e\left(\frac{2r}{175}\right) = 11.7266997655+-6.1224596863i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{175}(37,\cdot),\chi_{175}(1,\cdot)) = \sum_{r\in \Z/175\Z} \chi_{175}(37,r) \chi_{175}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{175}(37,·))
= \sum_{r \in \Z/175\Z}
\chi_{175}(37,r) e\left(\frac{1 r + 2 r^{-1}}{175}\right)
= 2.4081708325+-6.2734995015i \)