from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8100, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,14,0]))
pari: [g,chi] = znchar(Mod(4801,8100))
Basic properties
Modulus: | \(8100\) | |
Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(27\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{81}(22,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8100.bt
\(\chi_{8100}(301,\cdot)\) \(\chi_{8100}(601,\cdot)\) \(\chi_{8100}(1201,\cdot)\) \(\chi_{8100}(1501,\cdot)\) \(\chi_{8100}(2101,\cdot)\) \(\chi_{8100}(2401,\cdot)\) \(\chi_{8100}(3001,\cdot)\) \(\chi_{8100}(3301,\cdot)\) \(\chi_{8100}(3901,\cdot)\) \(\chi_{8100}(4201,\cdot)\) \(\chi_{8100}(4801,\cdot)\) \(\chi_{8100}(5101,\cdot)\) \(\chi_{8100}(5701,\cdot)\) \(\chi_{8100}(6001,\cdot)\) \(\chi_{8100}(6601,\cdot)\) \(\chi_{8100}(6901,\cdot)\) \(\chi_{8100}(7501,\cdot)\) \(\chi_{8100}(7801,\cdot)\)
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 27 polynomial |
Values on generators
\((4051,6401,7777)\) → \((1,e\left(\frac{7}{27}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 8100 }(4801, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{20}{27}\right)\) |
sage: chi.jacobi_sum(n)