sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(175, base_ring=CyclotomicField(60))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([21,50]))
pari: [g,chi] = znchar(Mod(103,175))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 175.x
\(\chi_{175}(3,\cdot)\) \(\chi_{175}(12,\cdot)\) \(\chi_{175}(17,\cdot)\) \(\chi_{175}(33,\cdot)\) \(\chi_{175}(38,\cdot)\) \(\chi_{175}(47,\cdot)\) \(\chi_{175}(52,\cdot)\) \(\chi_{175}(73,\cdot)\) \(\chi_{175}(87,\cdot)\) \(\chi_{175}(103,\cdot)\) \(\chi_{175}(108,\cdot)\) \(\chi_{175}(117,\cdot)\) \(\chi_{175}(122,\cdot)\) \(\chi_{175}(138,\cdot)\) \(\chi_{175}(152,\cdot)\) \(\chi_{175}(173,\cdot)\)
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
\((127,101)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{5}{6}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \(1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{175}(103,\cdot)) = \sum_{r\in \Z/175\Z} \chi_{175}(103,r) e\left(\frac{2r}{175}\right) = 6.3855957765+11.5855153782i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{175}(103,\cdot),\chi_{175}(1,\cdot)) = \sum_{r\in \Z/175\Z} \chi_{175}(103,r) \chi_{175}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{175}(103,·))
= \sum_{r \in \Z/175\Z}
\chi_{175}(103,r) e\left(\frac{1 r + 2 r^{-1}}{175}\right)
= -0.0 \)