from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2898, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,33,21]))
pari: [g,chi] = znchar(Mod(1903,2898))
Basic properties
| Modulus: | \(2898\) | |
| Conductor: | \(1449\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1449}(454,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2898.cp
\(\chi_{2898}(97,\cdot)\) \(\chi_{2898}(475,\cdot)\) \(\chi_{2898}(517,\cdot)\) \(\chi_{2898}(727,\cdot)\) \(\chi_{2898}(769,\cdot)\) \(\chi_{2898}(895,\cdot)\) \(\chi_{2898}(1147,\cdot)\) \(\chi_{2898}(1399,\cdot)\) \(\chi_{2898}(1483,\cdot)\) \(\chi_{2898}(1525,\cdot)\) \(\chi_{2898}(1735,\cdot)\) \(\chi_{2898}(1861,\cdot)\) \(\chi_{2898}(1903,\cdot)\) \(\chi_{2898}(2029,\cdot)\) \(\chi_{2898}(2113,\cdot)\) \(\chi_{2898}(2365,\cdot)\) \(\chi_{2898}(2407,\cdot)\) \(\chi_{2898}(2491,\cdot)\) \(\chi_{2898}(2659,\cdot)\) \(\chi_{2898}(2869,\cdot)\)
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 66 polynomial |
Values on generators
\((1289,829,1891)\) → \((e\left(\frac{1}{3}\right),-1,e\left(\frac{7}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2898 }(1903, a) \) | \(1\) | \(1\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{65}{66}\right)\) |
sage: chi.jacobi_sum(n)