from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7448, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,6,7]))
pari: [g,chi] = znchar(Mod(2003,7448))
Basic properties
| Modulus: | \(7448\) | |
| Conductor: | \(7448\) | 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 7448.ge
\(\chi_{7448}(715,\cdot)\) \(\chi_{7448}(939,\cdot)\) \(\chi_{7448}(1779,\cdot)\) \(\chi_{7448}(2003,\cdot)\) \(\chi_{7448}(3067,\cdot)\) \(\chi_{7448}(3907,\cdot)\) \(\chi_{7448}(4131,\cdot)\) \(\chi_{7448}(4971,\cdot)\) \(\chi_{7448}(6035,\cdot)\) \(\chi_{7448}(6259,\cdot)\) \(\chi_{7448}(7099,\cdot)\) \(\chi_{7448}(7323,\cdot)\)
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
\((1863,3725,3041,3137)\) → \((-1,-1,e\left(\frac{1}{7}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 7448 }(2003, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) |
sage: chi.jacobi_sum(n)