from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5148, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,30,36,55]))
pari: [g,chi] = znchar(Mod(449,5148))
Basic properties
| Modulus: | \(5148\) | |
| Conductor: | \(429\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{429}(20,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5148.iu
\(\chi_{5148}(449,\cdot)\) \(\chi_{5148}(665,\cdot)\) \(\chi_{5148}(773,\cdot)\) \(\chi_{5148}(917,\cdot)\) \(\chi_{5148}(1241,\cdot)\) \(\chi_{5148}(1709,\cdot)\) \(\chi_{5148}(1853,\cdot)\) \(\chi_{5148}(1961,\cdot)\) \(\chi_{5148}(2429,\cdot)\) \(\chi_{5148}(2645,\cdot)\) \(\chi_{5148}(2897,\cdot)\) \(\chi_{5148}(3833,\cdot)\) \(\chi_{5148}(3941,\cdot)\) \(\chi_{5148}(4409,\cdot)\) \(\chi_{5148}(4877,\cdot)\) \(\chi_{5148}(5129,\cdot)\)
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
\((2575,1145,937,4357)\) → \((1,-1,e\left(\frac{3}{5}\right),e\left(\frac{11}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 5148 }(449, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) |
sage: chi.jacobi_sum(n)