from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4608, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,39,32]))
pari: [g,chi] = znchar(Mod(97,4608))
Basic properties
| Modulus: | \(4608\) | |
| Conductor: | \(576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{576}(277,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4608.bq
\(\chi_{4608}(97,\cdot)\) \(\chi_{4608}(481,\cdot)\) \(\chi_{4608}(673,\cdot)\) \(\chi_{4608}(1057,\cdot)\) \(\chi_{4608}(1249,\cdot)\) \(\chi_{4608}(1633,\cdot)\) \(\chi_{4608}(1825,\cdot)\) \(\chi_{4608}(2209,\cdot)\) \(\chi_{4608}(2401,\cdot)\) \(\chi_{4608}(2785,\cdot)\) \(\chi_{4608}(2977,\cdot)\) \(\chi_{4608}(3361,\cdot)\) \(\chi_{4608}(3553,\cdot)\) \(\chi_{4608}(3937,\cdot)\) \(\chi_{4608}(4129,\cdot)\) \(\chi_{4608}(4513,\cdot)\)
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
\((3583,2053,4097)\) → \((1,e\left(\frac{13}{16}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 4608 }(97, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(-i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)