from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3344, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,12,40]))
pari: [g,chi] = znchar(Mod(125,3344))
Basic properties
| Modulus: | \(3344\) | |
| Conductor: | \(3344\) | 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 3344.eq
\(\chi_{3344}(125,\cdot)\) \(\chi_{3344}(581,\cdot)\) \(\chi_{3344}(653,\cdot)\) \(\chi_{3344}(885,\cdot)\) \(\chi_{3344}(1037,\cdot)\) \(\chi_{3344}(1109,\cdot)\) \(\chi_{3344}(1413,\cdot)\) \(\chi_{3344}(1565,\cdot)\) \(\chi_{3344}(1797,\cdot)\) \(\chi_{3344}(2253,\cdot)\) \(\chi_{3344}(2325,\cdot)\) \(\chi_{3344}(2557,\cdot)\) \(\chi_{3344}(2709,\cdot)\) \(\chi_{3344}(2781,\cdot)\) \(\chi_{3344}(3085,\cdot)\) \(\chi_{3344}(3237,\cdot)\)
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
\((2927,837,2433,705)\) → \((1,-i,e\left(\frac{1}{5}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 3344 }(125, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{30}\right)\) |
sage: chi.jacobi_sum(n)