from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4536, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,0,32,45]))
pari: [g,chi] = znchar(Mod(3127,4536))
Basic properties
| Modulus: | \(4536\) | |
| 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 \(\chi_{2268}(859,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4536.go
\(\chi_{4536}(103,\cdot)\) \(\chi_{4536}(367,\cdot)\) \(\chi_{4536}(607,\cdot)\) \(\chi_{4536}(871,\cdot)\) \(\chi_{4536}(1111,\cdot)\) \(\chi_{4536}(1375,\cdot)\) \(\chi_{4536}(1615,\cdot)\) \(\chi_{4536}(1879,\cdot)\) \(\chi_{4536}(2119,\cdot)\) \(\chi_{4536}(2383,\cdot)\) \(\chi_{4536}(2623,\cdot)\) \(\chi_{4536}(2887,\cdot)\) \(\chi_{4536}(3127,\cdot)\) \(\chi_{4536}(3391,\cdot)\) \(\chi_{4536}(3631,\cdot)\) \(\chi_{4536}(3895,\cdot)\) \(\chi_{4536}(4135,\cdot)\) \(\chi_{4536}(4399,\cdot)\)
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
\((1135,2269,3809,2593)\) → \((-1,1,e\left(\frac{16}{27}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 4536 }(3127, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) |
sage: chi.jacobi_sum(n)