sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(77, base_ring=CyclotomicField(30))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([10,21]))
pari: [g,chi] = znchar(Mod(51,77))
Basic properties
| Modulus: | \(77\) | |
| Conductor: | \(77\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 77.o
\(\chi_{77}(2,\cdot)\) \(\chi_{77}(18,\cdot)\) \(\chi_{77}(30,\cdot)\) \(\chi_{77}(39,\cdot)\) \(\chi_{77}(46,\cdot)\) \(\chi_{77}(51,\cdot)\) \(\chi_{77}(72,\cdot)\) \(\chi_{77}(74,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.0.1046076147688308987260717152173116396995512371.1 |
Values on generators
\((45,57)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{10}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \(-1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{10}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{77}(51,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(51,r) e\left(\frac{2r}{77}\right) = -3.6183988583+-7.9941972519i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{77}(51,\cdot),\chi_{77}(1,\cdot)) = \sum_{r\in \Z/77\Z} \chi_{77}(51,r) \chi_{77}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{77}(51,·))
= \sum_{r \in \Z/77\Z}
\chi_{77}(51,r) e\left(\frac{1 r + 2 r^{-1}}{77}\right)
= -2.5146560404+1.1195970029i \)