from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4600, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,0,33,32]))
pari: [g,chi] = znchar(Mod(4143,4600))
Basic properties
| Modulus: | \(4600\) | |
| 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 \(\chi_{460}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4600.cr
\(\chi_{4600}(407,\cdot)\) \(\chi_{4600}(607,\cdot)\) \(\chi_{4600}(807,\cdot)\) \(\chi_{4600}(1007,\cdot)\) \(\chi_{4600}(1143,\cdot)\) \(\chi_{4600}(1343,\cdot)\) \(\chi_{4600}(1407,\cdot)\) \(\chi_{4600}(1543,\cdot)\) \(\chi_{4600}(1743,\cdot)\) \(\chi_{4600}(1807,\cdot)\) \(\chi_{4600}(2007,\cdot)\) \(\chi_{4600}(2143,\cdot)\) \(\chi_{4600}(2543,\cdot)\) \(\chi_{4600}(2607,\cdot)\) \(\chi_{4600}(2743,\cdot)\) \(\chi_{4600}(3343,\cdot)\) \(\chi_{4600}(3407,\cdot)\) \(\chi_{4600}(3807,\cdot)\) \(\chi_{4600}(4143,\cdot)\) \(\chi_{4600}(4543,\cdot)\)
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
\((1151,2301,2577,1201)\) → \((-1,1,-i,e\left(\frac{8}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 4600 }(4143, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) |
sage: chi.jacobi_sum(n)