from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5148, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,50,24,45]))
pari: [g,chi] = znchar(Mod(5,5148))
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 \(\chi_{1287}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5148.ib
\(\chi_{5148}(5,\cdot)\) \(\chi_{5148}(317,\cdot)\) \(\chi_{5148}(785,\cdot)\) \(\chi_{5148}(905,\cdot)\) \(\chi_{5148}(1373,\cdot)\) \(\chi_{5148}(1721,\cdot)\) \(\chi_{5148}(1841,\cdot)\) \(\chi_{5148}(2621,\cdot)\) \(\chi_{5148}(2777,\cdot)\) \(\chi_{5148}(3089,\cdot)\) \(\chi_{5148}(3281,\cdot)\) \(\chi_{5148}(3557,\cdot)\) \(\chi_{5148}(3749,\cdot)\) \(\chi_{5148}(4217,\cdot)\) \(\chi_{5148}(4493,\cdot)\) \(\chi_{5148}(4997,\cdot)\)
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
\((2575,1145,937,4357)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{2}{5}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
\( \chi_{ 5148 }(5, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) |
sage: chi.jacobi_sum(n)