sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(648, base_ring=CyclotomicField(54))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([27,0,11]))
pari: [g,chi] = znchar(Mod(23,648))
Basic properties
| Modulus: | \(648\) | |
| Conductor: | \(324\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{324}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 648.be
\(\chi_{648}(23,\cdot)\) \(\chi_{648}(47,\cdot)\) \(\chi_{648}(95,\cdot)\) \(\chi_{648}(119,\cdot)\) \(\chi_{648}(167,\cdot)\) \(\chi_{648}(191,\cdot)\) \(\chi_{648}(239,\cdot)\) \(\chi_{648}(263,\cdot)\) \(\chi_{648}(311,\cdot)\) \(\chi_{648}(335,\cdot)\) \(\chi_{648}(383,\cdot)\) \(\chi_{648}(407,\cdot)\) \(\chi_{648}(455,\cdot)\) \(\chi_{648}(479,\cdot)\) \(\chi_{648}(527,\cdot)\) \(\chi_{648}(551,\cdot)\) \(\chi_{648}(599,\cdot)\) \(\chi_{648}(623,\cdot)\)
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
\((487,325,569)\) → \((-1,1,e\left(\frac{11}{54}\right))\)
Values
| \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \(1\) | \(1\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{31}{54}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 54 polynomial |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{648}(23,\cdot)) = \sum_{r\in \Z/648\Z} \chi_{648}(23,r) e\left(\frac{r}{324}\right) = -17.3920535747+-31.5200963269i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{648}(23,\cdot),\chi_{648}(1,\cdot)) = \sum_{r\in \Z/648\Z} \chi_{648}(23,r) \chi_{648}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{648}(23,·))
= \sum_{r \in \Z/648\Z}
\chi_{648}(23,r) e\left(\frac{1 r + 2 r^{-1}}{648}\right)
= -0.0 \)