from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,39,40]))
pari: [g,chi] = znchar(Mod(767,5600))
Basic properties
| Modulus: | \(5600\) | |
| Conductor: | \(700\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{700}(67,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5600.hj
\(\chi_{5600}(767,\cdot)\) \(\chi_{5600}(863,\cdot)\) \(\chi_{5600}(1087,\cdot)\) \(\chi_{5600}(1663,\cdot)\) \(\chi_{5600}(1887,\cdot)\) \(\chi_{5600}(1983,\cdot)\) \(\chi_{5600}(2783,\cdot)\) \(\chi_{5600}(3103,\cdot)\) \(\chi_{5600}(3327,\cdot)\) \(\chi_{5600}(3903,\cdot)\) \(\chi_{5600}(4127,\cdot)\) \(\chi_{5600}(4223,\cdot)\) \(\chi_{5600}(4447,\cdot)\) \(\chi_{5600}(5023,\cdot)\) \(\chi_{5600}(5247,\cdot)\) \(\chi_{5600}(5567,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((351,4901,5377,801)\) → \((-1,1,e\left(\frac{13}{20}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 5600 }(767, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{30}\right)\) |
sage: chi.jacobi_sum(n)