from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3700, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,21,20]))
pari: [g,chi] = znchar(Mod(1009,3700))
Basic properties
| Modulus: | \(3700\) | |
| Conductor: | \(925\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{925}(84,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3700.cr
\(\chi_{3700}(269,\cdot)\) \(\chi_{3700}(729,\cdot)\) \(\chi_{3700}(1009,\cdot)\) \(\chi_{3700}(1469,\cdot)\) \(\chi_{3700}(2209,\cdot)\) \(\chi_{3700}(2489,\cdot)\) \(\chi_{3700}(3229,\cdot)\) \(\chi_{3700}(3689,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((1851,1777,1001)\) → \((1,e\left(\frac{7}{10}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 3700 }(1009, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) |
sage: chi.jacobi_sum(n)