from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5700, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,1,0]))
pari: [g,chi] = znchar(Mod(1027,5700))
Basic properties
| Modulus: | \(5700\) | |
| Conductor: | \(100\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{100}(27,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5700.dj
\(\chi_{5700}(1027,\cdot)\) \(\chi_{5700}(1483,\cdot)\) \(\chi_{5700}(2167,\cdot)\) \(\chi_{5700}(2623,\cdot)\) \(\chi_{5700}(3763,\cdot)\) \(\chi_{5700}(4447,\cdot)\) \(\chi_{5700}(4903,\cdot)\) \(\chi_{5700}(5587,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | \(\Q(\zeta_{100})^+\) |
Values on generators
\((2851,1901,3877,4201)\) → \((-1,1,e\left(\frac{1}{20}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5700 }(1027, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)