from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2856, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,24,0,8,45]))
pari: [g,chi] = znchar(Mod(2845,2856))
Basic properties
| Modulus: | \(2856\) | |
| Conductor: | \(952\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{952}(941,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2856.fv
\(\chi_{2856}(61,\cdot)\) \(\chi_{2856}(397,\cdot)\) \(\chi_{2856}(997,\cdot)\) \(\chi_{2856}(1333,\cdot)\) \(\chi_{2856}(1405,\cdot)\) \(\chi_{2856}(1501,\cdot)\) \(\chi_{2856}(1669,\cdot)\) \(\chi_{2856}(1741,\cdot)\) \(\chi_{2856}(1909,\cdot)\) \(\chi_{2856}(2077,\cdot)\) \(\chi_{2856}(2173,\cdot)\) \(\chi_{2856}(2341,\cdot)\) \(\chi_{2856}(2509,\cdot)\) \(\chi_{2856}(2581,\cdot)\) \(\chi_{2856}(2749,\cdot)\) \(\chi_{2856}(2845,\cdot)\)
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
\((2143,1429,953,409,2689)\) → \((1,-1,1,e\left(\frac{1}{6}\right),e\left(\frac{15}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2856 }(2845, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(-i\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{13}{16}\right)\) |
sage: chi.jacobi_sum(n)