from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7448, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,10,21]))
pari: [g,chi] = znchar(Mod(151,7448))
Basic properties
| Modulus: | \(7448\) | |
| Conductor: | \(3724\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3724}(151,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7448.hk
\(\chi_{7448}(151,\cdot)\) \(\chi_{7448}(303,\cdot)\) \(\chi_{7448}(1215,\cdot)\) \(\chi_{7448}(1367,\cdot)\) \(\chi_{7448}(2279,\cdot)\) \(\chi_{7448}(3343,\cdot)\) \(\chi_{7448}(3495,\cdot)\) \(\chi_{7448}(4407,\cdot)\) \(\chi_{7448}(4559,\cdot)\) \(\chi_{7448}(5471,\cdot)\) \(\chi_{7448}(5623,\cdot)\) \(\chi_{7448}(6687,\cdot)\)
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
\((1863,3725,3041,3137)\) → \((-1,1,e\left(\frac{5}{21}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 7448 }(151, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) |
sage: chi.jacobi_sum(n)