from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6300, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,50,57,0]))
pari: [g,chi] = znchar(Mod(113,6300))
Basic properties
| Modulus: | \(6300\) | |
| 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 \(\chi_{225}(113,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6300.ip
\(\chi_{6300}(113,\cdot)\) \(\chi_{6300}(533,\cdot)\) \(\chi_{6300}(617,\cdot)\) \(\chi_{6300}(1037,\cdot)\) \(\chi_{6300}(1373,\cdot)\) \(\chi_{6300}(1877,\cdot)\) \(\chi_{6300}(2297,\cdot)\) \(\chi_{6300}(2633,\cdot)\) \(\chi_{6300}(3053,\cdot)\) \(\chi_{6300}(3137,\cdot)\) \(\chi_{6300}(4313,\cdot)\) \(\chi_{6300}(4397,\cdot)\) \(\chi_{6300}(4817,\cdot)\) \(\chi_{6300}(5153,\cdot)\) \(\chi_{6300}(5573,\cdot)\) \(\chi_{6300}(6077,\cdot)\)
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
\((3151,2801,3277,3601)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{19}{20}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6300 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)