from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4704, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,21,22]))
pari: [g,chi] = znchar(Mod(95,4704))
Basic properties
Modulus: | \(4704\) | |
Conductor: | \(588\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{588}(95,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4704.dp
\(\chi_{4704}(95,\cdot)\) \(\chi_{4704}(191,\cdot)\) \(\chi_{4704}(767,\cdot)\) \(\chi_{4704}(1535,\cdot)\) \(\chi_{4704}(2111,\cdot)\) \(\chi_{4704}(2207,\cdot)\) \(\chi_{4704}(2783,\cdot)\) \(\chi_{4704}(2879,\cdot)\) \(\chi_{4704}(3455,\cdot)\) \(\chi_{4704}(3551,\cdot)\) \(\chi_{4704}(4127,\cdot)\) \(\chi_{4704}(4223,\cdot)\)
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
\((1471,1765,3137,4609)\) → \((-1,1,-1,e\left(\frac{11}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 4704 }(95, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{16}{21}\right)\) |
sage: chi.jacobi_sum(n)