from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9800, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,21,1]))
pari: [g,chi] = znchar(Mod(199,9800))
Basic properties
| Modulus: | \(9800\) | |
| Conductor: | \(980\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{980}(199,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9800.ek
\(\chi_{9800}(199,\cdot)\) \(\chi_{9800}(2399,\cdot)\) \(\chi_{9800}(2999,\cdot)\) \(\chi_{9800}(3799,\cdot)\) \(\chi_{9800}(4399,\cdot)\) \(\chi_{9800}(5199,\cdot)\) \(\chi_{9800}(5799,\cdot)\) \(\chi_{9800}(6599,\cdot)\) \(\chi_{9800}(7199,\cdot)\) \(\chi_{9800}(7999,\cdot)\) \(\chi_{9800}(8599,\cdot)\) \(\chi_{9800}(9399,\cdot)\)
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: | 42.42.247844331230269810885249548811243543716772810770129718520393580847038464000000000000000000000.1 |
Values on generators
\((7351,4901,1177,5001)\) → \((-1,1,-1,e\left(\frac{1}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 9800 }(199, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)