from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8100, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,31,0]))
pari: [g,chi] = znchar(Mod(551,8100))
Basic properties
| Modulus: | \(8100\) | |
| 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 \(\chi_{324}(227,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8100.cj
\(\chi_{8100}(551,\cdot)\) \(\chi_{8100}(851,\cdot)\) \(\chi_{8100}(1451,\cdot)\) \(\chi_{8100}(1751,\cdot)\) \(\chi_{8100}(2351,\cdot)\) \(\chi_{8100}(2651,\cdot)\) \(\chi_{8100}(3251,\cdot)\) \(\chi_{8100}(3551,\cdot)\) \(\chi_{8100}(4151,\cdot)\) \(\chi_{8100}(4451,\cdot)\) \(\chi_{8100}(5051,\cdot)\) \(\chi_{8100}(5351,\cdot)\) \(\chi_{8100}(5951,\cdot)\) \(\chi_{8100}(6251,\cdot)\) \(\chi_{8100}(6851,\cdot)\) \(\chi_{8100}(7151,\cdot)\) \(\chi_{8100}(7751,\cdot)\) \(\chi_{8100}(8051,\cdot)\)
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
\((4051,6401,7777)\) → \((-1,e\left(\frac{31}{54}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8100 }(551, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{23}{54}\right)\) |
sage: chi.jacobi_sum(n)