from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2997, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([5,36]))
pari: [g,chi] = znchar(Mod(1490,2997))
Basic properties
| Modulus: | \(2997\) | |
| Conductor: | \(2997\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | 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 2997.ev
\(\chi_{2997}(137,\cdot)\) \(\chi_{2997}(158,\cdot)\) \(\chi_{2997}(470,\cdot)\) \(\chi_{2997}(491,\cdot)\) \(\chi_{2997}(803,\cdot)\) \(\chi_{2997}(824,\cdot)\) \(\chi_{2997}(1136,\cdot)\) \(\chi_{2997}(1157,\cdot)\) \(\chi_{2997}(1469,\cdot)\) \(\chi_{2997}(1490,\cdot)\) \(\chi_{2997}(1802,\cdot)\) \(\chi_{2997}(1823,\cdot)\) \(\chi_{2997}(2135,\cdot)\) \(\chi_{2997}(2156,\cdot)\) \(\chi_{2997}(2468,\cdot)\) \(\chi_{2997}(2489,\cdot)\) \(\chi_{2997}(2801,\cdot)\) \(\chi_{2997}(2822,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((1703,1297)\) → \((e\left(\frac{5}{54}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2997 }(1490, a) \) | \(-1\) | \(1\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) |
sage: chi.jacobi_sum(n)