from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5148, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,50,6,5]))
pari: [g,chi] = znchar(Mod(4721,5148))
Basic properties
Modulus: | \(5148\) | |
Conductor: | \(1287\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1287}(860,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5148.ip
\(\chi_{5148}(41,\cdot)\) \(\chi_{5148}(1073,\cdot)\) \(\chi_{5148}(1337,\cdot)\) \(\chi_{5148}(1865,\cdot)\) \(\chi_{5148}(2009,\cdot)\) \(\chi_{5148}(2273,\cdot)\) \(\chi_{5148}(2477,\cdot)\) \(\chi_{5148}(2741,\cdot)\) \(\chi_{5148}(2801,\cdot)\) \(\chi_{5148}(2945,\cdot)\) \(\chi_{5148}(3209,\cdot)\) \(\chi_{5148}(3269,\cdot)\) \(\chi_{5148}(3317,\cdot)\) \(\chi_{5148}(3737,\cdot)\) \(\chi_{5148}(4253,\cdot)\) \(\chi_{5148}(4721,\cdot)\)
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
\((2575,1145,937,4357)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{1}{10}\right),e\left(\frac{1}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
\( \chi_{ 5148 }(4721, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) |
sage: chi.jacobi_sum(n)