from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5635, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,2,0]))
pari: [g,chi] = znchar(Mod(254,5635))
Basic properties
| Modulus: | \(5635\) | |
| Conductor: | \(245\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{245}(9,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5635.bv
\(\chi_{5635}(254,\cdot)\) \(\chi_{5635}(599,\cdot)\) \(\chi_{5635}(1404,\cdot)\) \(\chi_{5635}(1864,\cdot)\) \(\chi_{5635}(2209,\cdot)\) \(\chi_{5635}(2669,\cdot)\) \(\chi_{5635}(3014,\cdot)\) \(\chi_{5635}(3474,\cdot)\) \(\chi_{5635}(3819,\cdot)\) \(\chi_{5635}(4279,\cdot)\) \(\chi_{5635}(5084,\cdot)\) \(\chi_{5635}(5429,\cdot)\)
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: | 42.42.8050468075656610214837511220114705524038488445061950919170859146595001220703125.1 |
Values on generators
\((3382,346,2696)\) → \((-1,e\left(\frac{1}{21}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 5635 }(254, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{20}{21}\right)\) |
sage: chi.jacobi_sum(n)