from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2205, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,21,26]))
pari: [g,chi] = znchar(Mod(2078,2205))
Basic properties
| Modulus: | \(2205\) | |
| Conductor: | \(735\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{735}(608,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2205.cx
\(\chi_{2205}(62,\cdot)\) \(\chi_{2205}(188,\cdot)\) \(\chi_{2205}(377,\cdot)\) \(\chi_{2205}(503,\cdot)\) \(\chi_{2205}(692,\cdot)\) \(\chi_{2205}(818,\cdot)\) \(\chi_{2205}(1007,\cdot)\) \(\chi_{2205}(1133,\cdot)\) \(\chi_{2205}(1448,\cdot)\) \(\chi_{2205}(1637,\cdot)\) \(\chi_{2205}(1952,\cdot)\) \(\chi_{2205}(2078,\cdot)\)
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: | 28.0.4101754449160695184473159618498838032884071911945819854736328125.1 |
Values on generators
\((1226,442,1081)\) → \((-1,-i,e\left(\frac{13}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 2205 }(2078, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) |
sage: chi.jacobi_sum(n)