from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5225, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([3,6,40]))
pari: [g,chi] = znchar(Mod(277,5225))
Basic properties
| Modulus: | \(5225\) | |
| Conductor: | \(5225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5225.gj
\(\chi_{5225}(277,\cdot)\) \(\chi_{5225}(558,\cdot)\) \(\chi_{5225}(767,\cdot)\) \(\chi_{5225}(1223,\cdot)\) \(\chi_{5225}(1337,\cdot)\) \(\chi_{5225}(1778,\cdot)\) \(\chi_{5225}(2063,\cdot)\) \(\chi_{5225}(2097,\cdot)\) \(\chi_{5225}(2477,\cdot)\) \(\chi_{5225}(3583,\cdot)\) \(\chi_{5225}(3792,\cdot)\) \(\chi_{5225}(3978,\cdot)\) \(\chi_{5225}(4248,\cdot)\) \(\chi_{5225}(4263,\cdot)\) \(\chi_{5225}(4362,\cdot)\) \(\chi_{5225}(5122,\cdot)\)
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
\((2927,2851,4676)\) → \((e\left(\frac{1}{20}\right),e\left(\frac{1}{10}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 5225 }(277, a) \) | \(1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) |
sage: chi.jacobi_sum(n)