from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1408, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,15,14]))
pari: [g,chi] = znchar(Mod(95,1408))
Basic properties
| Modulus: | \(1408\) | |
| Conductor: | \(176\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{176}(51,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1408.bf
\(\chi_{1408}(95,\cdot)\) \(\chi_{1408}(415,\cdot)\) \(\chi_{1408}(479,\cdot)\) \(\chi_{1408}(607,\cdot)\) \(\chi_{1408}(799,\cdot)\) \(\chi_{1408}(1119,\cdot)\) \(\chi_{1408}(1183,\cdot)\) \(\chi_{1408}(1311,\cdot)\)
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: | 20.20.200317132330035063121671003054276608.1 |
Values on generators
\((639,133,1025)\) → \((-1,-i,e\left(\frac{7}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 1408 }(95, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(i\) | \(1\) |
sage: chi.jacobi_sum(n)