from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(20160, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,9,16,12,32]))
pari: [g,chi] = znchar(Mod(67,20160))
Basic properties
Modulus: | \(20160\) | |
Conductor: | \(20160\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 20160.bat
\(\chi_{20160}(67,\cdot)\) \(\chi_{20160}(4363,\cdot)\) \(\chi_{20160}(4603,\cdot)\) \(\chi_{20160}(4867,\cdot)\) \(\chi_{20160}(5107,\cdot)\) \(\chi_{20160}(9403,\cdot)\) \(\chi_{20160}(9643,\cdot)\) \(\chi_{20160}(9907,\cdot)\) \(\chi_{20160}(10147,\cdot)\) \(\chi_{20160}(14443,\cdot)\) \(\chi_{20160}(14683,\cdot)\) \(\chi_{20160}(14947,\cdot)\) \(\chi_{20160}(15187,\cdot)\) \(\chi_{20160}(19483,\cdot)\) \(\chi_{20160}(19723,\cdot)\) \(\chi_{20160}(19987,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((8191,3781,17921,12097,8641)\) → \((-1,e\left(\frac{3}{16}\right),e\left(\frac{1}{3}\right),i,e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 20160 }(67, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{48}\right)\) |
sage: chi.jacobi_sum(n)