from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2898, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,55,45]))
pari: [g,chi] = znchar(Mod(19,2898))
Basic properties
| Modulus: | \(2898\) | |
| Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{161}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2898.cu
\(\chi_{2898}(19,\cdot)\) \(\chi_{2898}(145,\cdot)\) \(\chi_{2898}(199,\cdot)\) \(\chi_{2898}(451,\cdot)\) \(\chi_{2898}(523,\cdot)\) \(\chi_{2898}(649,\cdot)\) \(\chi_{2898}(1027,\cdot)\) \(\chi_{2898}(1207,\cdot)\) \(\chi_{2898}(1279,\cdot)\) \(\chi_{2898}(1459,\cdot)\) \(\chi_{2898}(1585,\cdot)\) \(\chi_{2898}(1837,\cdot)\) \(\chi_{2898}(2035,\cdot)\) \(\chi_{2898}(2089,\cdot)\) \(\chi_{2898}(2215,\cdot)\) \(\chi_{2898}(2287,\cdot)\) \(\chi_{2898}(2413,\cdot)\) \(\chi_{2898}(2593,\cdot)\) \(\chi_{2898}(2665,\cdot)\) \(\chi_{2898}(2719,\cdot)\)
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
\((1289,829,1891)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{15}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2898 }(19, a) \) | \(1\) | \(1\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) |
sage: chi.jacobi_sum(n)