from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,35,12]))
pari: [g,chi] = znchar(Mod(869,1764))
Basic properties
| Modulus: | \(1764\) | |
| Conductor: | \(441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{441}(428,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1764.ch
\(\chi_{1764}(29,\cdot)\) \(\chi_{1764}(113,\cdot)\) \(\chi_{1764}(281,\cdot)\) \(\chi_{1764}(365,\cdot)\) \(\chi_{1764}(533,\cdot)\) \(\chi_{1764}(617,\cdot)\) \(\chi_{1764}(869,\cdot)\) \(\chi_{1764}(1037,\cdot)\) \(\chi_{1764}(1121,\cdot)\) \(\chi_{1764}(1289,\cdot)\) \(\chi_{1764}(1541,\cdot)\) \(\chi_{1764}(1625,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((883,785,1081)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{2}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 1764 }(869, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)