from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4536, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,27,25,9]))
pari: [g,chi] = znchar(Mod(101,4536))
Basic properties
| Modulus: | \(4536\) | |
| Conductor: | \(4536\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4536.gs
\(\chi_{4536}(5,\cdot)\) \(\chi_{4536}(101,\cdot)\) \(\chi_{4536}(509,\cdot)\) \(\chi_{4536}(605,\cdot)\) \(\chi_{4536}(1013,\cdot)\) \(\chi_{4536}(1109,\cdot)\) \(\chi_{4536}(1517,\cdot)\) \(\chi_{4536}(1613,\cdot)\) \(\chi_{4536}(2021,\cdot)\) \(\chi_{4536}(2117,\cdot)\) \(\chi_{4536}(2525,\cdot)\) \(\chi_{4536}(2621,\cdot)\) \(\chi_{4536}(3029,\cdot)\) \(\chi_{4536}(3125,\cdot)\) \(\chi_{4536}(3533,\cdot)\) \(\chi_{4536}(3629,\cdot)\) \(\chi_{4536}(4037,\cdot)\) \(\chi_{4536}(4133,\cdot)\)
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
\((1135,2269,3809,2593)\) → \((1,-1,e\left(\frac{25}{54}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 4536 }(101, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{5}{18}\right)\) |
sage: chi.jacobi_sum(n)