from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3675, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,29]))
pari: [g,chi] = znchar(Mod(299,3675))
Basic properties
| Modulus: | \(3675\) | |
| Conductor: | \(735\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{735}(299,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3675.ck
\(\chi_{3675}(299,\cdot)\) \(\chi_{3675}(824,\cdot)\) \(\chi_{3675}(899,\cdot)\) \(\chi_{3675}(1349,\cdot)\) \(\chi_{3675}(1424,\cdot)\) \(\chi_{3675}(1874,\cdot)\) \(\chi_{3675}(1949,\cdot)\) \(\chi_{3675}(2399,\cdot)\) \(\chi_{3675}(2474,\cdot)\) \(\chi_{3675}(2924,\cdot)\) \(\chi_{3675}(2999,\cdot)\) \(\chi_{3675}(3524,\cdot)\)
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: | 42.42.589475176645907082922286550311127085690444572711075874815443834048428072939395904541015625.1 |
Values on generators
\((1226,1177,2551)\) → \((-1,-1,e\left(\frac{29}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 3675 }(299, a) \) | \(1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) |
sage: chi.jacobi_sum(n)