sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(196, base_ring=CyclotomicField(42))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,20]))
pari: [g,chi] = znchar(Mod(65,196))
Basic properties
| Modulus: | \(196\) | |
| Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{49}(16,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 196.m
\(\chi_{196}(9,\cdot)\) \(\chi_{196}(25,\cdot)\) \(\chi_{196}(37,\cdot)\) \(\chi_{196}(53,\cdot)\) \(\chi_{196}(65,\cdot)\) \(\chi_{196}(81,\cdot)\) \(\chi_{196}(93,\cdot)\) \(\chi_{196}(109,\cdot)\) \(\chi_{196}(121,\cdot)\) \(\chi_{196}(137,\cdot)\) \(\chi_{196}(149,\cdot)\) \(\chi_{196}(193,\cdot)\)
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: | \(\Q(\zeta_{49})^+\) |
Values on generators
\((99,101)\) → \((1,e\left(\frac{10}{21}\right))\)
Values
| \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \(1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{196}(65,\cdot)) = \sum_{r\in \Z/196\Z} \chi_{196}(65,r) e\left(\frac{r}{98}\right) = 8.9607695371+-10.7566077043i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{196}(65,\cdot),\chi_{196}(1,\cdot)) = \sum_{r\in \Z/196\Z} \chi_{196}(65,r) \chi_{196}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{196}(65,·))
= \sum_{r \in \Z/196\Z}
\chi_{196}(65,r) e\left(\frac{1 r + 2 r^{-1}}{196}\right)
= -0.0 \)