sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(101, base_ring=CyclotomicField(50))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([12]))
pari: [g,chi] = znchar(Mod(5,101))
Basic properties
| Modulus: | \(101\) | |
| Conductor: | \(101\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(25\) | 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 101.g
\(\chi_{101}(5,\cdot)\) \(\chi_{101}(16,\cdot)\) \(\chi_{101}(19,\cdot)\) \(\chi_{101}(24,\cdot)\) \(\chi_{101}(25,\cdot)\) \(\chi_{101}(31,\cdot)\) \(\chi_{101}(37,\cdot)\) \(\chi_{101}(52,\cdot)\) \(\chi_{101}(54,\cdot)\) \(\chi_{101}(56,\cdot)\) \(\chi_{101}(58,\cdot)\) \(\chi_{101}(68,\cdot)\) \(\chi_{101}(71,\cdot)\) \(\chi_{101}(78,\cdot)\) \(\chi_{101}(79,\cdot)\) \(\chi_{101}(80,\cdot)\) \(\chi_{101}(81,\cdot)\) \(\chi_{101}(88,\cdot)\) \(\chi_{101}(92,\cdot)\) \(\chi_{101}(97,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | 25.25.1269734648531914468903714880493455422104626762401.1 |
Values on generators
\(2\) → \(e\left(\frac{6}{25}\right)\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \(1\) | \(1\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(1\) | \(e\left(\frac{3}{25}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{101}(5,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(5,r) e\left(\frac{2r}{101}\right) = 7.8201713494+-6.3122832688i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{101}(5,\cdot),\chi_{101}(1,\cdot)) = \sum_{r\in \Z/101\Z} \chi_{101}(5,r) \chi_{101}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{101}(5,·))
= \sum_{r \in \Z/101\Z}
\chi_{101}(5,r) e\left(\frac{1 r + 2 r^{-1}}{101}\right)
= 9.6822257435+9.0922151686i \)