from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1323, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([7,15]))
pari: [g,chi] = znchar(Mod(629,1323))
Basic properties
Modulus: | \(1323\) | |
Conductor: | \(441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{441}(335,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1323.bt
\(\chi_{1323}(62,\cdot)\) \(\chi_{1323}(125,\cdot)\) \(\chi_{1323}(251,\cdot)\) \(\chi_{1323}(314,\cdot)\) \(\chi_{1323}(503,\cdot)\) \(\chi_{1323}(629,\cdot)\) \(\chi_{1323}(692,\cdot)\) \(\chi_{1323}(818,\cdot)\) \(\chi_{1323}(1007,\cdot)\) \(\chi_{1323}(1070,\cdot)\) \(\chi_{1323}(1196,\cdot)\) \(\chi_{1323}(1259,\cdot)\)
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
\((785,1081)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{5}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 1323 }(629, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)