from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9072, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,0,17,36]))
pari: [g,chi] = znchar(Mod(95,9072))
Basic properties
| Modulus: | \(9072\) | |
| Conductor: | \(2268\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2268}(95,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9072.is
\(\chi_{9072}(95,\cdot)\) \(\chi_{9072}(191,\cdot)\) \(\chi_{9072}(1103,\cdot)\) \(\chi_{9072}(1199,\cdot)\) \(\chi_{9072}(2111,\cdot)\) \(\chi_{9072}(2207,\cdot)\) \(\chi_{9072}(3119,\cdot)\) \(\chi_{9072}(3215,\cdot)\) \(\chi_{9072}(4127,\cdot)\) \(\chi_{9072}(4223,\cdot)\) \(\chi_{9072}(5135,\cdot)\) \(\chi_{9072}(5231,\cdot)\) \(\chi_{9072}(6143,\cdot)\) \(\chi_{9072}(6239,\cdot)\) \(\chi_{9072}(7151,\cdot)\) \(\chi_{9072}(7247,\cdot)\) \(\chi_{9072}(8159,\cdot)\) \(\chi_{9072}(8255,\cdot)\)
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
\((1135,6805,3809,2593)\) → \((-1,1,e\left(\frac{17}{54}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 9072 }(95, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{5}{9}\right)\) |
sage: chi.jacobi_sum(n)