from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,10,57]))
pari: [g,chi] = znchar(Mod(3413,3600))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3600.fv
\(\chi_{3600}(77,\cdot)\) \(\chi_{3600}(317,\cdot)\) \(\chi_{3600}(533,\cdot)\) \(\chi_{3600}(797,\cdot)\) \(\chi_{3600}(1013,\cdot)\) \(\chi_{3600}(1037,\cdot)\) \(\chi_{3600}(1253,\cdot)\) \(\chi_{3600}(1517,\cdot)\) \(\chi_{3600}(1733,\cdot)\) \(\chi_{3600}(1973,\cdot)\) \(\chi_{3600}(2237,\cdot)\) \(\chi_{3600}(2453,\cdot)\) \(\chi_{3600}(2477,\cdot)\) \(\chi_{3600}(3173,\cdot)\) \(\chi_{3600}(3197,\cdot)\) \(\chi_{3600}(3413,\cdot)\)
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
\((3151,901,2801,577)\) → \((1,i,e\left(\frac{1}{6}\right),e\left(\frac{19}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3600 }(3413, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) |
sage: chi.jacobi_sum(n)