from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1775, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([7,12]))
pari: [g,chi] = znchar(Mod(32,1775))
Basic properties
| Modulus: | \(1775\) | |
| Conductor: | \(355\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{355}(32,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1775.bu
\(\chi_{1775}(32,\cdot)\) \(\chi_{1775}(243,\cdot)\) \(\chi_{1775}(332,\cdot)\) \(\chi_{1775}(818,\cdot)\) \(\chi_{1775}(882,\cdot)\) \(\chi_{1775}(943,\cdot)\) \(\chi_{1775}(968,\cdot)\) \(\chi_{1775}(1168,\cdot)\) \(\chi_{1775}(1457,\cdot)\) \(\chi_{1775}(1468,\cdot)\) \(\chi_{1775}(1582,\cdot)\) \(\chi_{1775}(1607,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
| Fixed field: | 28.0.128401607345812283715591125636314028226206818103790283203125.1 |
Values on generators
\((427,1001)\) → \((i,e\left(\frac{3}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 1775 }(32, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) |
sage: chi.jacobi_sum(n)