from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5700, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,15,9,25]))
pari: [g,chi] = znchar(Mod(1589,5700))
Basic properties
| Modulus: | \(5700\) | |
| Conductor: | \(1425\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1425}(164,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5700.ds
\(\chi_{5700}(1589,\cdot)\) \(\chi_{5700}(1889,\cdot)\) \(\chi_{5700}(2729,\cdot)\) \(\chi_{5700}(3029,\cdot)\) \(\chi_{5700}(3869,\cdot)\) \(\chi_{5700}(4169,\cdot)\) \(\chi_{5700}(5009,\cdot)\) \(\chi_{5700}(5309,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.30.593101559206082927097940246428608574810414921785195474512875080108642578125.1 |
Values on generators
\((2851,1901,3877,4201)\) → \((1,-1,e\left(\frac{3}{10}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 5700 }(1589, a) \) | \(1\) | \(1\) | \(-1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)