sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(667, base_ring=CyclotomicField(28))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([14,27]))
pari: [g,chi] = znchar(Mod(160,667))
Basic properties
| Modulus: | \(667\) | |
| Conductor: | \(667\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | 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 667.o
\(\chi_{667}(68,\cdot)\) \(\chi_{667}(114,\cdot)\) \(\chi_{667}(137,\cdot)\) \(\chi_{667}(160,\cdot)\) \(\chi_{667}(206,\cdot)\) \(\chi_{667}(229,\cdot)\) \(\chi_{667}(275,\cdot)\) \(\chi_{667}(298,\cdot)\) \(\chi_{667}(321,\cdot)\) \(\chi_{667}(367,\cdot)\) \(\chi_{667}(482,\cdot)\) \(\chi_{667}(620,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | 28.28.35394489068231220324814698212289719250778220848093751207381.1 |
Values on generators
\((465,553)\) → \((-1,e\left(\frac{27}{28}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \(1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{667}(160,\cdot)) = \sum_{r\in \Z/667\Z} \chi_{667}(160,r) e\left(\frac{2r}{667}\right) = -8.9646667228+-24.2205439772i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{667}(160,\cdot),\chi_{667}(1,\cdot)) = \sum_{r\in \Z/667\Z} \chi_{667}(160,r) \chi_{667}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{667}(160,·))
= \sum_{r \in \Z/667\Z}
\chi_{667}(160,r) e\left(\frac{1 r + 2 r^{-1}}{667}\right)
= -1.2895879481+0.1453016654i \)