from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1950, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,1,10]))
pari: [g,chi] = znchar(Mod(77,1950))
Basic properties
| Modulus: | \(1950\) | |
| Conductor: | \(975\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{975}(77,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1950.bz
\(\chi_{1950}(77,\cdot)\) \(\chi_{1950}(233,\cdot)\) \(\chi_{1950}(467,\cdot)\) \(\chi_{1950}(623,\cdot)\) \(\chi_{1950}(1013,\cdot)\) \(\chi_{1950}(1247,\cdot)\) \(\chi_{1950}(1403,\cdot)\) \(\chi_{1950}(1637,\cdot)\)
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: | Number field defined by a degree 20 polynomial |
Values on generators
\((1301,1327,301)\) → \((-1,e\left(\frac{1}{20}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1950 }(77, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)