from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2205, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,21,26]))
pari: [g,chi] = znchar(Mod(118,2205))
Basic properties
| Modulus: | \(2205\) | |
| Conductor: | \(245\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{245}(118,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2205.cv
\(\chi_{2205}(118,\cdot)\) \(\chi_{2205}(307,\cdot)\) \(\chi_{2205}(433,\cdot)\) \(\chi_{2205}(622,\cdot)\) \(\chi_{2205}(748,\cdot)\) \(\chi_{2205}(937,\cdot)\) \(\chi_{2205}(1063,\cdot)\) \(\chi_{2205}(1252,\cdot)\) \(\chi_{2205}(1378,\cdot)\) \(\chi_{2205}(1693,\cdot)\) \(\chi_{2205}(1882,\cdot)\) \(\chi_{2205}(2197,\cdot)\)
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: | Number field defined by a degree 28 polynomial |
Values on generators
\((1226,442,1081)\) → \((1,-i,e\left(\frac{13}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 2205 }(118, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) |
sage: chi.jacobi_sum(n)