from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3724, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,1,14]))
pari: [g,chi] = znchar(Mod(1375,3724))
Basic properties
| Modulus: | \(3724\) | |
| Conductor: | \(3724\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | 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 3724.cy
\(\chi_{3724}(311,\cdot)\) \(\chi_{3724}(467,\cdot)\) \(\chi_{3724}(843,\cdot)\) \(\chi_{3724}(1375,\cdot)\) \(\chi_{3724}(1531,\cdot)\) \(\chi_{3724}(1907,\cdot)\) \(\chi_{3724}(2063,\cdot)\) \(\chi_{3724}(2439,\cdot)\) \(\chi_{3724}(2595,\cdot)\) \(\chi_{3724}(3127,\cdot)\) \(\chi_{3724}(3503,\cdot)\) \(\chi_{3724}(3659,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((1863,3041,3137)\) → \((-1,e\left(\frac{1}{42}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 3724 }(1375, a) \) | \(1\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) |
sage: chi.jacobi_sum(n)