from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2880, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,45,40,12]))
pari: [g,chi] = znchar(Mod(77,2880))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2880.fn
\(\chi_{2880}(77,\cdot)\) \(\chi_{2880}(293,\cdot)\) \(\chi_{2880}(317,\cdot)\) \(\chi_{2880}(533,\cdot)\) \(\chi_{2880}(797,\cdot)\) \(\chi_{2880}(1013,\cdot)\) \(\chi_{2880}(1037,\cdot)\) \(\chi_{2880}(1253,\cdot)\) \(\chi_{2880}(1517,\cdot)\) \(\chi_{2880}(1733,\cdot)\) \(\chi_{2880}(1757,\cdot)\) \(\chi_{2880}(1973,\cdot)\) \(\chi_{2880}(2237,\cdot)\) \(\chi_{2880}(2453,\cdot)\) \(\chi_{2880}(2477,\cdot)\) \(\chi_{2880}(2693,\cdot)\)
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
\((2431,901,641,577)\) → \((1,e\left(\frac{15}{16}\right),e\left(\frac{5}{6}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2880 }(77, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{24}\right)\) |
sage: chi.jacobi_sum(n)