from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1840, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,22,14]))
pari: [g,chi] = znchar(Mod(109,1840))
Basic properties
| Modulus: | \(1840\) | |
| Conductor: | \(1840\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1840.cu
\(\chi_{1840}(109,\cdot)\) \(\chi_{1840}(149,\cdot)\) \(\chi_{1840}(189,\cdot)\) \(\chi_{1840}(309,\cdot)\) \(\chi_{1840}(389,\cdot)\) \(\chi_{1840}(429,\cdot)\) \(\chi_{1840}(549,\cdot)\) \(\chi_{1840}(589,\cdot)\) \(\chi_{1840}(709,\cdot)\) \(\chi_{1840}(789,\cdot)\) \(\chi_{1840}(1029,\cdot)\) \(\chi_{1840}(1069,\cdot)\) \(\chi_{1840}(1109,\cdot)\) \(\chi_{1840}(1229,\cdot)\) \(\chi_{1840}(1309,\cdot)\) \(\chi_{1840}(1349,\cdot)\) \(\chi_{1840}(1469,\cdot)\) \(\chi_{1840}(1509,\cdot)\) \(\chi_{1840}(1629,\cdot)\) \(\chi_{1840}(1709,\cdot)\)
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: | Number field defined by a degree 44 polynomial |
Values on generators
\((1151,1381,737,1201)\) → \((1,-i,-1,e\left(\frac{7}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 1840 }(109, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) |
sage: chi.jacobi_sum(n)