from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5148, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,20,36,45]))
pari: [g,chi] = znchar(Mod(31,5148))
Basic properties
| Modulus: | \(5148\) | |
| Conductor: | \(5148\) | 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 5148.jd
\(\chi_{5148}(31,\cdot)\) \(\chi_{5148}(499,\cdot)\) \(\chi_{5148}(619,\cdot)\) \(\chi_{5148}(1087,\cdot)\) \(\chi_{5148}(1435,\cdot)\) \(\chi_{5148}(1555,\cdot)\) \(\chi_{5148}(2335,\cdot)\) \(\chi_{5148}(2491,\cdot)\) \(\chi_{5148}(2803,\cdot)\) \(\chi_{5148}(2995,\cdot)\) \(\chi_{5148}(3271,\cdot)\) \(\chi_{5148}(3463,\cdot)\) \(\chi_{5148}(3931,\cdot)\) \(\chi_{5148}(4207,\cdot)\) \(\chi_{5148}(4711,\cdot)\) \(\chi_{5148}(4867,\cdot)\)
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
\((2575,1145,937,4357)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{3}{5}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 5148 }(31, a) \) | \(1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) |
sage: chi.jacobi_sum(n)