from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2898, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,44,6]))
pari: [g,chi] = znchar(Mod(25,2898))
Basic properties
Modulus: | \(2898\) | |
Conductor: | \(1449\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1449}(25,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2898.bz
\(\chi_{2898}(25,\cdot)\) \(\chi_{2898}(121,\cdot)\) \(\chi_{2898}(151,\cdot)\) \(\chi_{2898}(403,\cdot)\) \(\chi_{2898}(499,\cdot)\) \(\chi_{2898}(625,\cdot)\) \(\chi_{2898}(877,\cdot)\) \(\chi_{2898}(1129,\cdot)\) \(\chi_{2898}(1159,\cdot)\) \(\chi_{2898}(1255,\cdot)\) \(\chi_{2898}(1411,\cdot)\) \(\chi_{2898}(1507,\cdot)\) \(\chi_{2898}(1789,\cdot)\) \(\chi_{2898}(1915,\cdot)\) \(\chi_{2898}(2263,\cdot)\) \(\chi_{2898}(2293,\cdot)\) \(\chi_{2898}(2419,\cdot)\) \(\chi_{2898}(2515,\cdot)\) \(\chi_{2898}(2671,\cdot)\) \(\chi_{2898}(2893,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 33 polynomial |
Values on generators
\((1289,829,1891)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{2}{3}\right),e\left(\frac{1}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 2898 }(25, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) |
sage: chi.jacobi_sum(n)