from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5166, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,10,21]))
pari: [g,chi] = znchar(Mod(185,5166))
Basic properties
| Modulus: | \(5166\) | |
| Conductor: | \(2583\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2583}(185,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5166.fn
\(\chi_{5166}(185,\cdot)\) \(\chi_{5166}(677,\cdot)\) \(\chi_{5166}(689,\cdot)\) \(\chi_{5166}(815,\cdot)\) \(\chi_{5166}(941,\cdot)\) \(\chi_{5166}(1181,\cdot)\) \(\chi_{5166}(1307,\cdot)\) \(\chi_{5166}(1433,\cdot)\) \(\chi_{5166}(2831,\cdot)\) \(\chi_{5166}(2957,\cdot)\) \(\chi_{5166}(3083,\cdot)\) \(\chi_{5166}(3323,\cdot)\) \(\chi_{5166}(3449,\cdot)\) \(\chi_{5166}(3575,\cdot)\) \(\chi_{5166}(3587,\cdot)\) \(\chi_{5166}(4079,\cdot)\)
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
\((2297,2215,3655)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{6}\right),e\left(\frac{7}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 5166 }(185, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) |
sage: chi.jacobi_sum(n)