from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8036, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,50,27]))
pari: [g,chi] = znchar(Mod(607,8036))
Basic properties
| Modulus: | \(8036\) | |
| Conductor: | \(1148\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1148}(607,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8036.df
\(\chi_{8036}(607,\cdot)\) \(\chi_{8036}(815,\cdot)\) \(\chi_{8036}(1783,\cdot)\) \(\chi_{8036}(2175,\cdot)\) \(\chi_{8036}(2383,\cdot)\) \(\chi_{8036}(3155,\cdot)\) \(\chi_{8036}(3547,\cdot)\) \(\chi_{8036}(3559,\cdot)\) \(\chi_{8036}(4723,\cdot)\) \(\chi_{8036}(4735,\cdot)\) \(\chi_{8036}(5127,\cdot)\) \(\chi_{8036}(5899,\cdot)\) \(\chi_{8036}(6107,\cdot)\) \(\chi_{8036}(6499,\cdot)\) \(\chi_{8036}(7467,\cdot)\) \(\chi_{8036}(7675,\cdot)\)
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
\((4019,493,785)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{9}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 8036 }(607, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) |
sage: chi.jacobi_sum(n)