from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,0,11,30]))
pari: [g,chi] = znchar(Mod(847,1840))
Basic properties
Modulus: | \(1840\) | |
Conductor: | \(460\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{460}(387,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1840.cp
\(\chi_{1840}(63,\cdot)\) \(\chi_{1840}(143,\cdot)\) \(\chi_{1840}(287,\cdot)\) \(\chi_{1840}(383,\cdot)\) \(\chi_{1840}(447,\cdot)\) \(\chi_{1840}(527,\cdot)\) \(\chi_{1840}(543,\cdot)\) \(\chi_{1840}(687,\cdot)\) \(\chi_{1840}(847,\cdot)\) \(\chi_{1840}(927,\cdot)\) \(\chi_{1840}(1023,\cdot)\) \(\chi_{1840}(1167,\cdot)\) \(\chi_{1840}(1183,\cdot)\) \(\chi_{1840}(1247,\cdot)\) \(\chi_{1840}(1263,\cdot)\) \(\chi_{1840}(1423,\cdot)\) \(\chi_{1840}(1487,\cdot)\) \(\chi_{1840}(1583,\cdot)\) \(\chi_{1840}(1647,\cdot)\) \(\chi_{1840}(1663,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
Fixed field: | 44.0.3190796191738142789235043789002363949895144644980550209800192000000000000000000000000000000000.1 |
Values on generators
\((1151,1381,737,1201)\) → \((-1,1,i,e\left(\frac{15}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 1840 }(847, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) |
sage: chi.jacobi_sum(n)