from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3700, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,12,35]))
pari: [g,chi] = znchar(Mod(991,3700))
Basic properties
| Modulus: | \(3700\) | |
| Conductor: | \(3700\) | 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 3700.dv
\(\chi_{3700}(171,\cdot)\) \(\chi_{3700}(711,\cdot)\) \(\chi_{3700}(791,\cdot)\) \(\chi_{3700}(911,\cdot)\) \(\chi_{3700}(991,\cdot)\) \(\chi_{3700}(1531,\cdot)\) \(\chi_{3700}(1731,\cdot)\) \(\chi_{3700}(2191,\cdot)\) \(\chi_{3700}(2271,\cdot)\) \(\chi_{3700}(2391,\cdot)\) \(\chi_{3700}(2471,\cdot)\) \(\chi_{3700}(2931,\cdot)\) \(\chi_{3700}(3011,\cdot)\) \(\chi_{3700}(3131,\cdot)\) \(\chi_{3700}(3211,\cdot)\) \(\chi_{3700}(3671,\cdot)\)
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
\((1851,1777,1001)\) → \((-1,e\left(\frac{1}{5}\right),e\left(\frac{7}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 3700 }(991, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) |
sage: chi.jacobi_sum(n)