from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5148, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,20,36,35]))
pari: [g,chi] = znchar(Mod(427,5148))
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.io
\(\chi_{5148}(427,\cdot)\) \(\chi_{5148}(895,\cdot)\) \(\chi_{5148}(1411,\cdot)\) \(\chi_{5148}(1831,\cdot)\) \(\chi_{5148}(1879,\cdot)\) \(\chi_{5148}(1939,\cdot)\) \(\chi_{5148}(2203,\cdot)\) \(\chi_{5148}(2347,\cdot)\) \(\chi_{5148}(2407,\cdot)\) \(\chi_{5148}(2671,\cdot)\) \(\chi_{5148}(2875,\cdot)\) \(\chi_{5148}(3139,\cdot)\) \(\chi_{5148}(3283,\cdot)\) \(\chi_{5148}(3811,\cdot)\) \(\chi_{5148}(4075,\cdot)\) \(\chi_{5148}(5107,\cdot)\)
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
\((2575,1145,937,4357)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{3}{5}\right),e\left(\frac{7}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
\( \chi_{ 5148 }(427, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) |
sage: chi.jacobi_sum(n)