from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6300, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,20,57,0]))
pari: [g,chi] = znchar(Mod(463,6300))
Basic properties
| Modulus: | \(6300\) | |
| Conductor: | \(900\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{900}(463,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6300.ie
\(\chi_{6300}(463,\cdot)\) \(\chi_{6300}(547,\cdot)\) \(\chi_{6300}(967,\cdot)\) \(\chi_{6300}(1303,\cdot)\) \(\chi_{6300}(1723,\cdot)\) \(\chi_{6300}(2227,\cdot)\) \(\chi_{6300}(2563,\cdot)\) \(\chi_{6300}(2983,\cdot)\) \(\chi_{6300}(3067,\cdot)\) \(\chi_{6300}(3487,\cdot)\) \(\chi_{6300}(3823,\cdot)\) \(\chi_{6300}(4327,\cdot)\) \(\chi_{6300}(4747,\cdot)\) \(\chi_{6300}(5083,\cdot)\) \(\chi_{6300}(5503,\cdot)\) \(\chi_{6300}(5587,\cdot)\)
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
\((3151,2801,3277,3601)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{19}{20}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6300 }(463, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)