from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4275, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,33,10]))
pari: [g,chi] = znchar(Mod(2098,4275))
Basic properties
| Modulus: | \(4275\) | |
| Conductor: | \(475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{475}(198,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4275.fo
\(\chi_{4275}(217,\cdot)\) \(\chi_{4275}(388,\cdot)\) \(\chi_{4275}(487,\cdot)\) \(\chi_{4275}(658,\cdot)\) \(\chi_{4275}(1072,\cdot)\) \(\chi_{4275}(1342,\cdot)\) \(\chi_{4275}(1513,\cdot)\) \(\chi_{4275}(1927,\cdot)\) \(\chi_{4275}(2098,\cdot)\) \(\chi_{4275}(2197,\cdot)\) \(\chi_{4275}(2953,\cdot)\) \(\chi_{4275}(3052,\cdot)\) \(\chi_{4275}(3223,\cdot)\) \(\chi_{4275}(3637,\cdot)\) \(\chi_{4275}(3808,\cdot)\) \(\chi_{4275}(4078,\cdot)\)
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
\((1901,1027,1351)\) → \((1,e\left(\frac{11}{20}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
| \( \chi_{ 4275 }(2098, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(-i\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) |
sage: chi.jacobi_sum(n)