from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(38025, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,51,40]))
pari: [g,chi] = znchar(Mod(22,38025))
Basic properties
| Modulus: | \(38025\) | |
| Conductor: | \(2925\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2925}(22,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 38025.ia
\(\chi_{38025}(22,\cdot)\) \(\chi_{38025}(5047,\cdot)\) \(\chi_{38025}(7627,\cdot)\) \(\chi_{38025}(9148,\cdot)\) \(\chi_{38025}(12652,\cdot)\) \(\chi_{38025}(14173,\cdot)\) \(\chi_{38025}(16753,\cdot)\) \(\chi_{38025}(21778,\cdot)\) \(\chi_{38025}(22837,\cdot)\) \(\chi_{38025}(24358,\cdot)\) \(\chi_{38025}(27862,\cdot)\) \(\chi_{38025}(29383,\cdot)\) \(\chi_{38025}(30442,\cdot)\) \(\chi_{38025}(31963,\cdot)\) \(\chi_{38025}(35467,\cdot)\) \(\chi_{38025}(36988,\cdot)\)
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
\((29576,9127,37351)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{17}{20}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
| \( \chi_{ 38025 }(22, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) |
sage: chi.jacobi_sum(n)