from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1122, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,12,35]))
pari: [g,chi] = znchar(Mod(19,1122))
Basic properties
| Modulus: | \(1122\) | |
| Conductor: | \(187\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{187}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1122.bh
\(\chi_{1122}(19,\cdot)\) \(\chi_{1122}(127,\cdot)\) \(\chi_{1122}(145,\cdot)\) \(\chi_{1122}(151,\cdot)\) \(\chi_{1122}(325,\cdot)\) \(\chi_{1122}(349,\cdot)\) \(\chi_{1122}(457,\cdot)\) \(\chi_{1122}(535,\cdot)\) \(\chi_{1122}(655,\cdot)\) \(\chi_{1122}(733,\cdot)\) \(\chi_{1122}(739,\cdot)\) \(\chi_{1122}(865,\cdot)\) \(\chi_{1122}(937,\cdot)\) \(\chi_{1122}(943,\cdot)\) \(\chi_{1122}(1063,\cdot)\) \(\chi_{1122}(1069,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.0.359624204259227998212313764863527746816862563620018205460931204658277030572367073.1 |
Values on generators
\((749,409,1057)\) → \((1,e\left(\frac{3}{10}\right),e\left(\frac{7}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 1122 }(19, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{40}\right)\) |
sage: chi.jacobi_sum(n)