from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1950, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,3,50]))
pari: [g,chi] = znchar(Mod(127,1950))
Basic properties
| Modulus: | \(1950\) | |
| Conductor: | \(325\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{325}(127,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1950.ct
\(\chi_{1950}(127,\cdot)\) \(\chi_{1950}(277,\cdot)\) \(\chi_{1950}(283,\cdot)\) \(\chi_{1950}(433,\cdot)\) \(\chi_{1950}(517,\cdot)\) \(\chi_{1950}(667,\cdot)\) \(\chi_{1950}(673,\cdot)\) \(\chi_{1950}(823,\cdot)\) \(\chi_{1950}(1063,\cdot)\) \(\chi_{1950}(1213,\cdot)\) \(\chi_{1950}(1297,\cdot)\) \(\chi_{1950}(1447,\cdot)\) \(\chi_{1950}(1453,\cdot)\) \(\chi_{1950}(1603,\cdot)\) \(\chi_{1950}(1687,\cdot)\) \(\chi_{1950}(1837,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((1301,1327,301)\) → \((1,e\left(\frac{1}{20}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1950 }(127, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)