from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4725, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,27,20]))
pari: [g,chi] = znchar(Mod(37,4725))
Basic properties
| Modulus: | \(4725\) | |
| Conductor: | \(1575\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1575}(562,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4725.gb
\(\chi_{4725}(37,\cdot)\) \(\chi_{4725}(613,\cdot)\) \(\chi_{4725}(802,\cdot)\) \(\chi_{4725}(1558,\cdot)\) \(\chi_{4725}(1738,\cdot)\) \(\chi_{4725}(1747,\cdot)\) \(\chi_{4725}(1927,\cdot)\) \(\chi_{4725}(2503,\cdot)\) \(\chi_{4725}(2683,\cdot)\) \(\chi_{4725}(2692,\cdot)\) \(\chi_{4725}(2872,\cdot)\) \(\chi_{4725}(3448,\cdot)\) \(\chi_{4725}(3628,\cdot)\) \(\chi_{4725}(3637,\cdot)\) \(\chi_{4725}(3817,\cdot)\) \(\chi_{4725}(4573,\cdot)\)
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
\((4376,1702,2026)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{9}{20}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 4725 }(37, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) |
sage: chi.jacobi_sum(n)