from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2880, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,21,8,36]))
pari: [g,chi] = znchar(Mod(83,2880))
Basic properties
| Modulus: | \(2880\) | |
| Conductor: | \(2880\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2880.ez
\(\chi_{2880}(83,\cdot)\) \(\chi_{2880}(347,\cdot)\) \(\chi_{2880}(563,\cdot)\) \(\chi_{2880}(587,\cdot)\) \(\chi_{2880}(803,\cdot)\) \(\chi_{2880}(1067,\cdot)\) \(\chi_{2880}(1283,\cdot)\) \(\chi_{2880}(1307,\cdot)\) \(\chi_{2880}(1523,\cdot)\) \(\chi_{2880}(1787,\cdot)\) \(\chi_{2880}(2003,\cdot)\) \(\chi_{2880}(2027,\cdot)\) \(\chi_{2880}(2243,\cdot)\) \(\chi_{2880}(2507,\cdot)\) \(\chi_{2880}(2723,\cdot)\) \(\chi_{2880}(2747,\cdot)\)
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
\((2431,901,641,577)\) → \((-1,e\left(\frac{7}{16}\right),e\left(\frac{1}{6}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2880 }(83, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(-1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{23}{24}\right)\) |
sage: chi.jacobi_sum(n)