sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(147, base_ring=CyclotomicField(14))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,6]))
pari: [g,chi] = znchar(Mod(127,147))
Basic properties
| Modulus: | \(147\) | |
| Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(7\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{49}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 147.i
\(\chi_{147}(22,\cdot)\) \(\chi_{147}(43,\cdot)\) \(\chi_{147}(64,\cdot)\) \(\chi_{147}(85,\cdot)\) \(\chi_{147}(106,\cdot)\) \(\chi_{147}(127,\cdot)\)
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: | \(\Q(\zeta_{7})\) |
| Fixed field: | 7.7.13841287201.1 |
Values on generators
\((50,52)\) → \((1,e\left(\frac{3}{7}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \(1\) | \(1\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(1\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{147}(127,\cdot)) = \sum_{r\in \Z/147\Z} \chi_{147}(127,r) e\left(\frac{2r}{147}\right) = 4.0048166209+5.7412057822i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{147}(127,\cdot),\chi_{147}(1,\cdot)) = \sum_{r\in \Z/147\Z} \chi_{147}(127,r) \chi_{147}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{147}(127,·))
= \sum_{r \in \Z/147\Z}
\chi_{147}(127,r) e\left(\frac{1 r + 2 r^{-1}}{147}\right)
= 0.0 \)