from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1032, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,0,19]))
pari: [g,chi] = znchar(Mod(19,1032))
Basic properties
| Modulus: | \(1032\) | |
| Conductor: | \(344\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{344}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1032.cg
\(\chi_{1032}(19,\cdot)\) \(\chi_{1032}(91,\cdot)\) \(\chi_{1032}(115,\cdot)\) \(\chi_{1032}(163,\cdot)\) \(\chi_{1032}(235,\cdot)\) \(\chi_{1032}(331,\cdot)\) \(\chi_{1032}(499,\cdot)\) \(\chi_{1032}(571,\cdot)\) \(\chi_{1032}(691,\cdot)\) \(\chi_{1032}(835,\cdot)\) \(\chi_{1032}(931,\cdot)\) \(\chi_{1032}(979,\cdot)\)
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: | 42.42.86515994746897550947675385197225985831622982825258543026271873735800731799783414431744.1 |
Values on generators
\((775,517,689,433)\) → \((-1,-1,1,e\left(\frac{19}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1032 }(19, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) |
sage: chi.jacobi_sum(n)