from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,20,27]))
pari: [g,chi] = znchar(Mod(337,3600))
Basic properties
Modulus: | \(3600\) | |
Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{225}(112,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3600.fy
\(\chi_{3600}(97,\cdot)\) \(\chi_{3600}(337,\cdot)\) \(\chi_{3600}(673,\cdot)\) \(\chi_{3600}(817,\cdot)\) \(\chi_{3600}(913,\cdot)\) \(\chi_{3600}(1537,\cdot)\) \(\chi_{3600}(1633,\cdot)\) \(\chi_{3600}(1777,\cdot)\) \(\chi_{3600}(2113,\cdot)\) \(\chi_{3600}(2353,\cdot)\) \(\chi_{3600}(2497,\cdot)\) \(\chi_{3600}(2833,\cdot)\) \(\chi_{3600}(2977,\cdot)\) \(\chi_{3600}(3073,\cdot)\) \(\chi_{3600}(3217,\cdot)\) \(\chi_{3600}(3553,\cdot)\)
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
\((3151,901,2801,577)\) → \((1,1,e\left(\frac{1}{3}\right),e\left(\frac{9}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3600 }(337, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) |
sage: chi.jacobi_sum(n)