from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2592, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,27,7]))
pari: [g,chi] = znchar(Mod(47,2592))
Basic properties
| Modulus: | \(2592\) | |
| Conductor: | \(648\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{648}(371,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2592.bv
\(\chi_{2592}(47,\cdot)\) \(\chi_{2592}(239,\cdot)\) \(\chi_{2592}(335,\cdot)\) \(\chi_{2592}(527,\cdot)\) \(\chi_{2592}(623,\cdot)\) \(\chi_{2592}(815,\cdot)\) \(\chi_{2592}(911,\cdot)\) \(\chi_{2592}(1103,\cdot)\) \(\chi_{2592}(1199,\cdot)\) \(\chi_{2592}(1391,\cdot)\) \(\chi_{2592}(1487,\cdot)\) \(\chi_{2592}(1679,\cdot)\) \(\chi_{2592}(1775,\cdot)\) \(\chi_{2592}(1967,\cdot)\) \(\chi_{2592}(2063,\cdot)\) \(\chi_{2592}(2255,\cdot)\) \(\chi_{2592}(2351,\cdot)\) \(\chi_{2592}(2543,\cdot)\)
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
\((2431,325,1217)\) → \((-1,-1,e\left(\frac{7}{54}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 2592 }(47, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{5}{54}\right)\) |
sage: chi.jacobi_sum(n)