sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(99, base_ring=CyclotomicField(30))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([20,18]))
pari: [g,chi] = znchar(Mod(97,99))
Basic properties
| Modulus: | \(99\) | |
| Conductor: | \(99\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(15\) | 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 99.m
\(\chi_{99}(4,\cdot)\) \(\chi_{99}(16,\cdot)\) \(\chi_{99}(25,\cdot)\) \(\chi_{99}(31,\cdot)\) \(\chi_{99}(49,\cdot)\) \(\chi_{99}(58,\cdot)\) \(\chi_{99}(70,\cdot)\) \(\chi_{99}(97,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 15.15.10943023107606534329121.1 |
Values on generators
\((56,46)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{3}{5}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{99}(97,\cdot)) = \sum_{r\in \Z/99\Z} \chi_{99}(97,r) e\left(\frac{2r}{99}\right) = -0.2393067105+-9.9469961445i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{99}(97,\cdot),\chi_{99}(1,\cdot)) = \sum_{r\in \Z/99\Z} \chi_{99}(97,r) \chi_{99}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{99}(97,·))
= \sum_{r \in \Z/99\Z}
\chi_{99}(97,r) e\left(\frac{1 r + 2 r^{-1}}{99}\right)
= 8.6228160033+9.5766073664i \)