from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2303, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([26,21]))
pari: [g,chi] = znchar(Mod(1080,2303))
Basic properties
Modulus: | \(2303\) | |
Conductor: | \(2303\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2303.q
\(\chi_{2303}(46,\cdot)\) \(\chi_{2303}(93,\cdot)\) \(\chi_{2303}(375,\cdot)\) \(\chi_{2303}(751,\cdot)\) \(\chi_{2303}(1033,\cdot)\) \(\chi_{2303}(1080,\cdot)\) \(\chi_{2303}(1362,\cdot)\) \(\chi_{2303}(1409,\cdot)\) \(\chi_{2303}(1691,\cdot)\) \(\chi_{2303}(1738,\cdot)\) \(\chi_{2303}(2020,\cdot)\) \(\chi_{2303}(2067,\cdot)\)
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
\((2257,99)\) → \((e\left(\frac{13}{21}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
\( \chi_{ 2303 }(1080, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) |
sage: chi.jacobi_sum(n)