from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,7,2]))
pari: [g,chi] = znchar(Mod(695,1764))
Basic properties
| Modulus: | \(1764\) | |
| Conductor: | \(1764\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1764.cz
\(\chi_{1764}(95,\cdot)\) \(\chi_{1764}(191,\cdot)\) \(\chi_{1764}(347,\cdot)\) \(\chi_{1764}(443,\cdot)\) \(\chi_{1764}(599,\cdot)\) \(\chi_{1764}(695,\cdot)\) \(\chi_{1764}(947,\cdot)\) \(\chi_{1764}(1103,\cdot)\) \(\chi_{1764}(1199,\cdot)\) \(\chi_{1764}(1355,\cdot)\) \(\chi_{1764}(1607,\cdot)\) \(\chi_{1764}(1703,\cdot)\)
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: | 42.42.84986492613867422664202192058457487324646368884457377881199611944629462212222730322047304527187202914910208.2 |
Values on generators
\((883,785,1081)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{1}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 1764 }(695, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{21}\right)\) |
sage: chi.jacobi_sum(n)