sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(429, base_ring=CyclotomicField(30))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([15,21,10]))
pari: [g,chi] = znchar(Mod(29,429))
Basic properties
| Modulus: | \(429\) | |
| Conductor: | \(429\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 429.bq
\(\chi_{429}(29,\cdot)\) \(\chi_{429}(35,\cdot)\) \(\chi_{429}(68,\cdot)\) \(\chi_{429}(74,\cdot)\) \(\chi_{429}(107,\cdot)\) \(\chi_{429}(347,\cdot)\) \(\chi_{429}(380,\cdot)\) \(\chi_{429}(425,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Values on generators
\((287,79,67)\) → \((-1,e\left(\frac{7}{10}\right),e\left(\frac{1}{3}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.30.3575101417044962746290924719176231718719784535246431948097.1 |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{429}(29,\cdot)) = \sum_{r\in \Z/429\Z} \chi_{429}(29,r) e\left(\frac{2r}{429}\right) = -4.0880559007+20.3048713109i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{429}(29,\cdot),\chi_{429}(1,\cdot)) = \sum_{r\in \Z/429\Z} \chi_{429}(29,r) \chi_{429}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{429}(29,·))
= \sum_{r \in \Z/429\Z}
\chi_{429}(29,r) e\left(\frac{1 r + 2 r^{-1}}{429}\right)
= 0.0 \)