from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,50,24]))
pari: [g,chi] = znchar(Mod(2381,3600))
Basic properties
Modulus: | \(3600\) | |
Conductor: | \(3600\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3600.gd
\(\chi_{3600}(221,\cdot)\) \(\chi_{3600}(461,\cdot)\) \(\chi_{3600}(581,\cdot)\) \(\chi_{3600}(821,\cdot)\) \(\chi_{3600}(941,\cdot)\) \(\chi_{3600}(1181,\cdot)\) \(\chi_{3600}(1541,\cdot)\) \(\chi_{3600}(1661,\cdot)\) \(\chi_{3600}(2021,\cdot)\) \(\chi_{3600}(2261,\cdot)\) \(\chi_{3600}(2381,\cdot)\) \(\chi_{3600}(2621,\cdot)\) \(\chi_{3600}(2741,\cdot)\) \(\chi_{3600}(2981,\cdot)\) \(\chi_{3600}(3341,\cdot)\) \(\chi_{3600}(3461,\cdot)\)
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
\((3151,901,2801,577)\) → \((1,-i,e\left(\frac{5}{6}\right),e\left(\frac{2}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3600 }(2381, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) |
sage: chi.jacobi_sum(n)