from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5700, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,10,1,0]))
pari: [g,chi] = znchar(Mod(77,5700))
Basic properties
| Modulus: | \(5700\) | |
| Conductor: | \(75\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{75}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5700.dh
\(\chi_{5700}(77,\cdot)\) \(\chi_{5700}(533,\cdot)\) \(\chi_{5700}(1217,\cdot)\) \(\chi_{5700}(1673,\cdot)\) \(\chi_{5700}(2813,\cdot)\) \(\chi_{5700}(3497,\cdot)\) \(\chi_{5700}(3953,\cdot)\) \(\chi_{5700}(4637,\cdot)\)
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: | \(\Q(\zeta_{75})^+\) |
Values on generators
\((2851,1901,3877,4201)\) → \((1,-1,e\left(\frac{1}{20}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5700 }(77, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)