from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2808, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,0,8,15]))
pari: [g,chi] = znchar(Mod(97,2808))
Basic properties
| Modulus: | \(2808\) | |
| Conductor: | \(351\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{351}(97,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2808.hf
\(\chi_{2808}(97,\cdot)\) \(\chi_{2808}(193,\cdot)\) \(\chi_{2808}(241,\cdot)\) \(\chi_{2808}(769,\cdot)\) \(\chi_{2808}(1033,\cdot)\) \(\chi_{2808}(1129,\cdot)\) \(\chi_{2808}(1177,\cdot)\) \(\chi_{2808}(1705,\cdot)\) \(\chi_{2808}(1969,\cdot)\) \(\chi_{2808}(2065,\cdot)\) \(\chi_{2808}(2113,\cdot)\) \(\chi_{2808}(2641,\cdot)\)
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: | Number field defined by a degree 36 polynomial |
Values on generators
\((703,1405,2081,1081)\) → \((1,1,e\left(\frac{2}{9}\right),e\left(\frac{5}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2808 }(97, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-i\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)