sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(93, base_ring=CyclotomicField(30))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,16]))
pari: [g,chi] = znchar(Mod(28,93))
Basic properties
| Modulus: | \(93\) | |
| Conductor: | \(31\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{31}(28,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 93.m
\(\chi_{93}(7,\cdot)\) \(\chi_{93}(10,\cdot)\) \(\chi_{93}(19,\cdot)\) \(\chi_{93}(28,\cdot)\) \(\chi_{93}(40,\cdot)\) \(\chi_{93}(49,\cdot)\) \(\chi_{93}(76,\cdot)\) \(\chi_{93}(82,\cdot)\)
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: | \(\Q(\zeta_{31})^+\) |
Values on generators
\((32,34)\) → \((1,e\left(\frac{8}{15}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{93}(28,\cdot)) = \sum_{r\in \Z/93\Z} \chi_{93}(28,r) e\left(\frac{2r}{93}\right) = 4.5067515285+-3.2694327735i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{93}(28,\cdot),\chi_{93}(1,\cdot)) = \sum_{r\in \Z/93\Z} \chi_{93}(28,r) \chi_{93}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{93}(28,·))
= \sum_{r \in \Z/93\Z}
\chi_{93}(28,r) e\left(\frac{1 r + 2 r^{-1}}{93}\right)
= 14.0698306906+-10.2223303586i \)