from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1444, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,5]))
pari: [g,chi] = znchar(Mod(37,1444))
Basic properties
| Modulus: | \(1444\) | |
| Conductor: | \(361\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{361}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1444.o
\(\chi_{1444}(37,\cdot)\) \(\chi_{1444}(113,\cdot)\) \(\chi_{1444}(189,\cdot)\) \(\chi_{1444}(265,\cdot)\) \(\chi_{1444}(341,\cdot)\) \(\chi_{1444}(417,\cdot)\) \(\chi_{1444}(493,\cdot)\) \(\chi_{1444}(569,\cdot)\) \(\chi_{1444}(645,\cdot)\) \(\chi_{1444}(797,\cdot)\) \(\chi_{1444}(873,\cdot)\) \(\chi_{1444}(949,\cdot)\) \(\chi_{1444}(1025,\cdot)\) \(\chi_{1444}(1101,\cdot)\) \(\chi_{1444}(1177,\cdot)\) \(\chi_{1444}(1253,\cdot)\) \(\chi_{1444}(1329,\cdot)\) \(\chi_{1444}(1405,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
| Fixed field: | 38.0.2233638411813024816853081773648251688534529753590642239923912316757382599022775822751448518259.1 |
Values on generators
\((723,1085)\) → \((1,e\left(\frac{5}{38}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 1444 }(37, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) |
sage: chi.jacobi_sum(n)