from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1224, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,0,32,3]))
pari: [g,chi] = znchar(Mod(241,1224))
Basic properties
Modulus: | \(1224\) | |
Conductor: | \(153\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{153}(88,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1224.db
\(\chi_{1224}(97,\cdot)\) \(\chi_{1224}(193,\cdot)\) \(\chi_{1224}(241,\cdot)\) \(\chi_{1224}(265,\cdot)\) \(\chi_{1224}(313,\cdot)\) \(\chi_{1224}(337,\cdot)\) \(\chi_{1224}(385,\cdot)\) \(\chi_{1224}(481,\cdot)\) \(\chi_{1224}(601,\cdot)\) \(\chi_{1224}(673,\cdot)\) \(\chi_{1224}(745,\cdot)\) \(\chi_{1224}(889,\cdot)\) \(\chi_{1224}(913,\cdot)\) \(\chi_{1224}(1057,\cdot)\) \(\chi_{1224}(1129,\cdot)\) \(\chi_{1224}(1201,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((919,613,137,649)\) → \((1,1,e\left(\frac{2}{3}\right),e\left(\frac{1}{16}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 1224 }(241, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)