from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2808, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,18,4,3]))
pari: [g,chi] = znchar(Mod(301,2808))
Basic properties
| Modulus: | \(2808\) | |
| Conductor: | \(2808\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | 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 2808.gy
\(\chi_{2808}(301,\cdot)\) \(\chi_{2808}(565,\cdot)\) \(\chi_{2808}(661,\cdot)\) \(\chi_{2808}(709,\cdot)\) \(\chi_{2808}(1237,\cdot)\) \(\chi_{2808}(1501,\cdot)\) \(\chi_{2808}(1597,\cdot)\) \(\chi_{2808}(1645,\cdot)\) \(\chi_{2808}(2173,\cdot)\) \(\chi_{2808}(2437,\cdot)\) \(\chi_{2808}(2533,\cdot)\) \(\chi_{2808}(2581,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.0.100558967911072114670827528216731307434473269211199849804559244243297483058318543570163174735872.1 |
Values on generators
\((703,1405,2081,1081)\) → \((1,-1,e\left(\frac{1}{9}\right),e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2808 }(301, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(i\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)