from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2880, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,3,32,36]))
pari: [g,chi] = znchar(Mod(133,2880))
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.fa
\(\chi_{2880}(133,\cdot)\) \(\chi_{2880}(157,\cdot)\) \(\chi_{2880}(373,\cdot)\) \(\chi_{2880}(637,\cdot)\) \(\chi_{2880}(853,\cdot)\) \(\chi_{2880}(877,\cdot)\) \(\chi_{2880}(1093,\cdot)\) \(\chi_{2880}(1357,\cdot)\) \(\chi_{2880}(1573,\cdot)\) \(\chi_{2880}(1597,\cdot)\) \(\chi_{2880}(1813,\cdot)\) \(\chi_{2880}(2077,\cdot)\) \(\chi_{2880}(2293,\cdot)\) \(\chi_{2880}(2317,\cdot)\) \(\chi_{2880}(2533,\cdot)\) \(\chi_{2880}(2797,\cdot)\)
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
\((2431,901,641,577)\) → \((1,e\left(\frac{1}{16}\right),e\left(\frac{2}{3}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2880 }(133, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(-1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{5}{24}\right)\) |
sage: chi.jacobi_sum(n)