from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4600, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,0,33,6]))
pari: [g,chi] = znchar(Mod(2793,4600))
Basic properties
| Modulus: | \(4600\) | |
| Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{115}(33,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4600.ct
\(\chi_{4600}(57,\cdot)\) \(\chi_{4600}(457,\cdot)\) \(\chi_{4600}(793,\cdot)\) \(\chi_{4600}(1193,\cdot)\) \(\chi_{4600}(1257,\cdot)\) \(\chi_{4600}(1857,\cdot)\) \(\chi_{4600}(1993,\cdot)\) \(\chi_{4600}(2057,\cdot)\) \(\chi_{4600}(2457,\cdot)\) \(\chi_{4600}(2593,\cdot)\) \(\chi_{4600}(2793,\cdot)\) \(\chi_{4600}(2857,\cdot)\) \(\chi_{4600}(3057,\cdot)\) \(\chi_{4600}(3193,\cdot)\) \(\chi_{4600}(3257,\cdot)\) \(\chi_{4600}(3457,\cdot)\) \(\chi_{4600}(3593,\cdot)\) \(\chi_{4600}(3793,\cdot)\) \(\chi_{4600}(3993,\cdot)\) \(\chi_{4600}(4193,\cdot)\)
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: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((1151,2301,2577,1201)\) → \((1,1,-i,e\left(\frac{3}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 4600 }(2793, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) |
sage: chi.jacobi_sum(n)