from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2835, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([5,27,36]))
pari: [g,chi] = znchar(Mod(599,2835))
Basic properties
| Modulus: | \(2835\) | |
| Conductor: | \(2835\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2835.ei
\(\chi_{2835}(254,\cdot)\) \(\chi_{2835}(284,\cdot)\) \(\chi_{2835}(569,\cdot)\) \(\chi_{2835}(599,\cdot)\) \(\chi_{2835}(884,\cdot)\) \(\chi_{2835}(914,\cdot)\) \(\chi_{2835}(1199,\cdot)\) \(\chi_{2835}(1229,\cdot)\) \(\chi_{2835}(1514,\cdot)\) \(\chi_{2835}(1544,\cdot)\) \(\chi_{2835}(1829,\cdot)\) \(\chi_{2835}(1859,\cdot)\) \(\chi_{2835}(2144,\cdot)\) \(\chi_{2835}(2174,\cdot)\) \(\chi_{2835}(2459,\cdot)\) \(\chi_{2835}(2489,\cdot)\) \(\chi_{2835}(2774,\cdot)\) \(\chi_{2835}(2804,\cdot)\)
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
\((1541,1702,2026)\) → \((e\left(\frac{5}{54}\right),-1,e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 2835 }(599, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) |
sage: chi.jacobi_sum(n)