from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4600, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,22,33,28]))
pari: [g,chi] = znchar(Mod(243,4600))
Basic properties
| Modulus: | \(4600\) | |
| Conductor: | \(920\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{920}(243,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4600.co
\(\chi_{4600}(243,\cdot)\) \(\chi_{4600}(307,\cdot)\) \(\chi_{4600}(443,\cdot)\) \(\chi_{4600}(1043,\cdot)\) \(\chi_{4600}(1107,\cdot)\) \(\chi_{4600}(1507,\cdot)\) \(\chi_{4600}(1843,\cdot)\) \(\chi_{4600}(2243,\cdot)\) \(\chi_{4600}(2707,\cdot)\) \(\chi_{4600}(2907,\cdot)\) \(\chi_{4600}(3107,\cdot)\) \(\chi_{4600}(3307,\cdot)\) \(\chi_{4600}(3443,\cdot)\) \(\chi_{4600}(3643,\cdot)\) \(\chi_{4600}(3707,\cdot)\) \(\chi_{4600}(3843,\cdot)\) \(\chi_{4600}(4043,\cdot)\) \(\chi_{4600}(4107,\cdot)\) \(\chi_{4600}(4307,\cdot)\) \(\chi_{4600}(4443,\cdot)\)
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
\((1151,2301,2577,1201)\) → \((-1,-1,-i,e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 4600 }(243, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) |
sage: chi.jacobi_sum(n)