sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(147, base_ring=CyclotomicField(42))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([21,19]))
pari: [g,chi] = znchar(Mod(38,147))
Basic properties
| Modulus: | \(147\) | |
| Conductor: | \(147\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | 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 147.o
\(\chi_{147}(5,\cdot)\) \(\chi_{147}(17,\cdot)\) \(\chi_{147}(26,\cdot)\) \(\chi_{147}(38,\cdot)\) \(\chi_{147}(47,\cdot)\) \(\chi_{147}(59,\cdot)\) \(\chi_{147}(89,\cdot)\) \(\chi_{147}(101,\cdot)\) \(\chi_{147}(110,\cdot)\) \(\chi_{147}(122,\cdot)\) \(\chi_{147}(131,\cdot)\) \(\chi_{147}(143,\cdot)\)
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: | \(\Q(\zeta_{147})^+\) |
Values on generators
\((50,52)\) → \((-1,e\left(\frac{19}{42}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \(1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{147}(38,\cdot)) = \sum_{r\in \Z/147\Z} \chi_{147}(38,r) e\left(\frac{2r}{147}\right) = -6.3956047422+-10.3003029072i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{147}(38,\cdot),\chi_{147}(1,\cdot)) = \sum_{r\in \Z/147\Z} \chi_{147}(38,r) \chi_{147}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{147}(38,·))
= \sum_{r \in \Z/147\Z}
\chi_{147}(38,r) e\left(\frac{1 r + 2 r^{-1}}{147}\right)
= 0.0 \)