from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1075, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([27,10]))
pari: [g,chi] = znchar(Mod(37,1075))
Basic properties
| Modulus: | \(1075\) | |
| Conductor: | \(1075\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1075.bi
\(\chi_{1075}(37,\cdot)\) \(\chi_{1075}(123,\cdot)\) \(\chi_{1075}(222,\cdot)\) \(\chi_{1075}(252,\cdot)\) \(\chi_{1075}(308,\cdot)\) \(\chi_{1075}(338,\cdot)\) \(\chi_{1075}(437,\cdot)\) \(\chi_{1075}(467,\cdot)\) \(\chi_{1075}(523,\cdot)\) \(\chi_{1075}(553,\cdot)\) \(\chi_{1075}(652,\cdot)\) \(\chi_{1075}(738,\cdot)\) \(\chi_{1075}(867,\cdot)\) \(\chi_{1075}(897,\cdot)\) \(\chi_{1075}(953,\cdot)\) \(\chi_{1075}(983,\cdot)\)
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
\((302,476)\) → \((e\left(\frac{9}{20}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 1075 }(37, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) |
sage: chi.jacobi_sum(n)