from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,19]))
pari: [g,chi] = znchar(Mod(577,1764))
Basic properties
Modulus: | \(1764\) | |
Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{49}(38,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1764.co
\(\chi_{1764}(73,\cdot)\) \(\chi_{1764}(145,\cdot)\) \(\chi_{1764}(397,\cdot)\) \(\chi_{1764}(577,\cdot)\) \(\chi_{1764}(649,\cdot)\) \(\chi_{1764}(829,\cdot)\) \(\chi_{1764}(1081,\cdot)\) \(\chi_{1764}(1153,\cdot)\) \(\chi_{1764}(1333,\cdot)\) \(\chi_{1764}(1405,\cdot)\) \(\chi_{1764}(1585,\cdot)\) \(\chi_{1764}(1657,\cdot)\)
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
\((883,785,1081)\) → \((1,1,e\left(\frac{19}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 1764 }(577, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{10}{21}\right)\) |
sage: chi.jacobi_sum(n)