from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,20,39]))
pari: [g,chi] = znchar(Mod(2317,3600))
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.fu
\(\chi_{3600}(133,\cdot)\) \(\chi_{3600}(373,\cdot)\) \(\chi_{3600}(637,\cdot)\) \(\chi_{3600}(853,\cdot)\) \(\chi_{3600}(877,\cdot)\) \(\chi_{3600}(1573,\cdot)\) \(\chi_{3600}(1597,\cdot)\) \(\chi_{3600}(1813,\cdot)\) \(\chi_{3600}(2077,\cdot)\) \(\chi_{3600}(2317,\cdot)\) \(\chi_{3600}(2533,\cdot)\) \(\chi_{3600}(2797,\cdot)\) \(\chi_{3600}(3013,\cdot)\) \(\chi_{3600}(3037,\cdot)\) \(\chi_{3600}(3253,\cdot)\) \(\chi_{3600}(3517,\cdot)\)
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
\((3151,901,2801,577)\) → \((1,-i,e\left(\frac{1}{3}\right),e\left(\frac{13}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3600 }(2317, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{23}{30}\right)\) |
sage: chi.jacobi_sum(n)