from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5700, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,57,40]))
pari: [g,chi] = znchar(Mod(163,5700))
Basic properties
| Modulus: | \(5700\) | |
| Conductor: | \(1900\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1900}(163,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5700.eo
\(\chi_{5700}(163,\cdot)\) \(\chi_{5700}(463,\cdot)\) \(\chi_{5700}(847,\cdot)\) \(\chi_{5700}(1147,\cdot)\) \(\chi_{5700}(1303,\cdot)\) \(\chi_{5700}(1603,\cdot)\) \(\chi_{5700}(1987,\cdot)\) \(\chi_{5700}(2287,\cdot)\) \(\chi_{5700}(3127,\cdot)\) \(\chi_{5700}(3427,\cdot)\) \(\chi_{5700}(3583,\cdot)\) \(\chi_{5700}(3883,\cdot)\) \(\chi_{5700}(4267,\cdot)\) \(\chi_{5700}(4567,\cdot)\) \(\chi_{5700}(4723,\cdot)\) \(\chi_{5700}(5023,\cdot)\)
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
\((2851,1901,3877,4201)\) → \((-1,1,e\left(\frac{19}{20}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5700 }(163, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)