from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6027, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,10,9]))
pari: [g,chi] = znchar(Mod(80,6027))
Basic properties
| Modulus: | \(6027\) | |
| Conductor: | \(861\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{861}(80,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6027.df
\(\chi_{6027}(80,\cdot)\) \(\chi_{6027}(374,\cdot)\) \(\chi_{6027}(815,\cdot)\) \(\chi_{6027}(1109,\cdot)\) \(\chi_{6027}(1538,\cdot)\) \(\chi_{6027}(1550,\cdot)\) \(\chi_{6027}(2714,\cdot)\) \(\chi_{6027}(2726,\cdot)\) \(\chi_{6027}(3155,\cdot)\) \(\chi_{6027}(3449,\cdot)\) \(\chi_{6027}(3890,\cdot)\) \(\chi_{6027}(4184,\cdot)\) \(\chi_{6027}(4490,\cdot)\) \(\chi_{6027}(4625,\cdot)\) \(\chi_{6027}(5666,\cdot)\) \(\chi_{6027}(5801,\cdot)\)
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
\((4019,493,2794)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{3}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 6027 }(80, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{11}{60}\right)\) |
sage: chi.jacobi_sum(n)