from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1824, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,3,12,8]))
pari: [g,chi] = znchar(Mod(197,1824))
Basic properties
| Modulus: | \(1824\) | |
| Conductor: | \(1824\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | 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 1824.cw
\(\chi_{1824}(125,\cdot)\) \(\chi_{1824}(197,\cdot)\) \(\chi_{1824}(581,\cdot)\) \(\chi_{1824}(653,\cdot)\) \(\chi_{1824}(1037,\cdot)\) \(\chi_{1824}(1109,\cdot)\) \(\chi_{1824}(1493,\cdot)\) \(\chi_{1824}(1565,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | Number field defined by a degree 24 polynomial |
Values on generators
\((799,229,1217,97)\) → \((1,e\left(\frac{1}{8}\right),-1,e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1824 }(197, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{24}\right)\) | \(i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(1\) | \(e\left(\frac{5}{24}\right)\) |
sage: chi.jacobi_sum(n)