from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2793, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,32,21]))
pari: [g,chi] = znchar(Mod(37,2793))
Basic properties
| Modulus: | \(2793\) | |
| Conductor: | \(931\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{931}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2793.dl
\(\chi_{2793}(37,\cdot)\) \(\chi_{2793}(151,\cdot)\) \(\chi_{2793}(436,\cdot)\) \(\chi_{2793}(550,\cdot)\) \(\chi_{2793}(835,\cdot)\) \(\chi_{2793}(1234,\cdot)\) \(\chi_{2793}(1348,\cdot)\) \(\chi_{2793}(1633,\cdot)\) \(\chi_{2793}(1747,\cdot)\) \(\chi_{2793}(2032,\cdot)\) \(\chi_{2793}(2146,\cdot)\) \(\chi_{2793}(2545,\cdot)\)
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: | Number field defined by a degree 42 polynomial |
Values on generators
\((932,2110,2206)\) → \((1,e\left(\frac{16}{21}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(20\) |
| \( \chi_{ 2793 }(37, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) |
sage: chi.jacobi_sum(n)