from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1305, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,21,10]))
pari: [g,chi] = znchar(Mod(1198,1305))
Basic properties
| Modulus: | \(1305\) | |
| Conductor: | \(145\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{145}(38,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1305.cd
\(\chi_{1305}(208,\cdot)\) \(\chi_{1305}(352,\cdot)\) \(\chi_{1305}(613,\cdot)\) \(\chi_{1305}(622,\cdot)\) \(\chi_{1305}(847,\cdot)\) \(\chi_{1305}(883,\cdot)\) \(\chi_{1305}(892,\cdot)\) \(\chi_{1305}(937,\cdot)\) \(\chi_{1305}(1108,\cdot)\) \(\chi_{1305}(1153,\cdot)\) \(\chi_{1305}(1198,\cdot)\) \(\chi_{1305}(1252,\cdot)\)
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: | 28.0.50201655190081835380839261671426578388690948486328125.1 |
Values on generators
\((146,262,901)\) → \((1,-i,e\left(\frac{5}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1305 }(1198, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(i\) | \(e\left(\frac{5}{7}\right)\) |
sage: chi.jacobi_sum(n)