sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(129, base_ring=CyclotomicField(42))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,10]))
pari: [g,chi] = znchar(Mod(10,129))
Basic properties
| Modulus: | \(129\) | |
| Conductor: | \(43\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{43}(10,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 129.m
\(\chi_{129}(10,\cdot)\) \(\chi_{129}(13,\cdot)\) \(\chi_{129}(25,\cdot)\) \(\chi_{129}(31,\cdot)\) \(\chi_{129}(40,\cdot)\) \(\chi_{129}(52,\cdot)\) \(\chi_{129}(58,\cdot)\) \(\chi_{129}(67,\cdot)\) \(\chi_{129}(100,\cdot)\) \(\chi_{129}(103,\cdot)\) \(\chi_{129}(109,\cdot)\) \(\chi_{129}(124,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | \(\Q(\zeta_{43})^+\) |
Values on generators
\((44,46)\) → \((1,e\left(\frac{5}{21}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \(1\) | \(1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{129}(10,\cdot)) = \sum_{r\in \Z/129\Z} \chi_{129}(10,r) e\left(\frac{2r}{129}\right) = -6.3450292498+-1.6554769159i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{129}(10,\cdot),\chi_{129}(1,\cdot)) = \sum_{r\in \Z/129\Z} \chi_{129}(10,r) \chi_{129}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{129}(10,·))
= \sum_{r \in \Z/129\Z}
\chi_{129}(10,r) e\left(\frac{1 r + 2 r^{-1}}{129}\right)
= 3.092810724+13.550489153i \)