from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5184, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,27,37]))
pari: [g,chi] = znchar(Mod(2783,5184))
Basic properties
| Modulus: | \(5184\) | |
| 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 \(\chi_{648}(515,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5184.cd
\(\chi_{5184}(95,\cdot)\) \(\chi_{5184}(479,\cdot)\) \(\chi_{5184}(671,\cdot)\) \(\chi_{5184}(1055,\cdot)\) \(\chi_{5184}(1247,\cdot)\) \(\chi_{5184}(1631,\cdot)\) \(\chi_{5184}(1823,\cdot)\) \(\chi_{5184}(2207,\cdot)\) \(\chi_{5184}(2399,\cdot)\) \(\chi_{5184}(2783,\cdot)\) \(\chi_{5184}(2975,\cdot)\) \(\chi_{5184}(3359,\cdot)\) \(\chi_{5184}(3551,\cdot)\) \(\chi_{5184}(3935,\cdot)\) \(\chi_{5184}(4127,\cdot)\) \(\chi_{5184}(4511,\cdot)\) \(\chi_{5184}(4703,\cdot)\) \(\chi_{5184}(5087,\cdot)\)
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
\((2431,325,1217)\) → \((-1,-1,e\left(\frac{37}{54}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 5184 }(2783, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{11}{54}\right)\) |
sage: chi.jacobi_sum(n)