from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3888, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,0,16]))
pari: [g,chi] = znchar(Mod(2287,3888))
Basic properties
| Modulus: | \(3888\) | |
| Conductor: | \(324\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{324}(7,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3888.bm
\(\chi_{3888}(127,\cdot)\) \(\chi_{3888}(415,\cdot)\) \(\chi_{3888}(559,\cdot)\) \(\chi_{3888}(847,\cdot)\) \(\chi_{3888}(991,\cdot)\) \(\chi_{3888}(1279,\cdot)\) \(\chi_{3888}(1423,\cdot)\) \(\chi_{3888}(1711,\cdot)\) \(\chi_{3888}(1855,\cdot)\) \(\chi_{3888}(2143,\cdot)\) \(\chi_{3888}(2287,\cdot)\) \(\chi_{3888}(2575,\cdot)\) \(\chi_{3888}(2719,\cdot)\) \(\chi_{3888}(3007,\cdot)\) \(\chi_{3888}(3151,\cdot)\) \(\chi_{3888}(3439,\cdot)\) \(\chi_{3888}(3583,\cdot)\) \(\chi_{3888}(3871,\cdot)\)
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
\((2431,2917,1217)\) → \((-1,1,e\left(\frac{8}{27}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 3888 }(2287, a) \) | \(-1\) | \(1\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{23}{54}\right)\) |
sage: chi.jacobi_sum(n)