from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9216, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,39,16]))
pari: [g,chi] = znchar(Mod(193,9216))
Basic properties
| Modulus: | \(9216\) | |
| 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 \(\chi_{576}(85,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9216.bq
\(\chi_{9216}(193,\cdot)\) \(\chi_{9216}(961,\cdot)\) \(\chi_{9216}(1345,\cdot)\) \(\chi_{9216}(2113,\cdot)\) \(\chi_{9216}(2497,\cdot)\) \(\chi_{9216}(3265,\cdot)\) \(\chi_{9216}(3649,\cdot)\) \(\chi_{9216}(4417,\cdot)\) \(\chi_{9216}(4801,\cdot)\) \(\chi_{9216}(5569,\cdot)\) \(\chi_{9216}(5953,\cdot)\) \(\chi_{9216}(6721,\cdot)\) \(\chi_{9216}(7105,\cdot)\) \(\chi_{9216}(7873,\cdot)\) \(\chi_{9216}(8257,\cdot)\) \(\chi_{9216}(9025,\cdot)\)
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
\((8191,2053,4097)\) → \((1,e\left(\frac{13}{16}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 9216 }(193, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(-i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)