from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4026, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,12,19]))
pari: [g,chi] = znchar(Mod(49,4026))
Basic properties
| Modulus: | \(4026\) | |
| Conductor: | \(671\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{671}(49,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4026.fj
\(\chi_{4026}(49,\cdot)\) \(\chi_{4026}(493,\cdot)\) \(\chi_{4026}(751,\cdot)\) \(\chi_{4026}(829,\cdot)\) \(\chi_{4026}(1015,\cdot)\) \(\chi_{4026}(1285,\cdot)\) \(\chi_{4026}(1753,\cdot)\) \(\chi_{4026}(2425,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.30.58612420827331088167192282070112190732220663165316857394692862412334982504781.4 |
Values on generators
\((1343,1465,3235)\) → \((1,e\left(\frac{2}{5}\right),e\left(\frac{19}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4026 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) |
sage: chi.jacobi_sum(n)