from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3021, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,4,27]))
pari: [g,chi] = znchar(Mod(23,3021))
Basic properties
| Modulus: | \(3021\) | |
| Conductor: | \(3021\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | 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 3021.bo
\(\chi_{3021}(23,\cdot)\) \(\chi_{3021}(500,\cdot)\) \(\chi_{3021}(560,\cdot)\) \(\chi_{3021}(719,\cdot)\) \(\chi_{3021}(878,\cdot)\) \(\chi_{3021}(1355,\cdot)\) \(\chi_{3021}(1613,\cdot)\) \(\chi_{3021}(1772,\cdot)\) \(\chi_{3021}(2468,\cdot)\) \(\chi_{3021}(2627,\cdot)\) \(\chi_{3021}(2726,\cdot)\) \(\chi_{3021}(2885,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((2015,1750,2281)\) → \((-1,e\left(\frac{1}{9}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 3021 }(23, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) |
sage: chi.jacobi_sum(n)