from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(861, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,20,21]))
pari: [g,chi] = znchar(Mod(76,861))
Basic properties
| Modulus: | \(861\) | |
| Conductor: | \(287\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{287}(76,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 861.cd
\(\chi_{861}(13,\cdot)\) \(\chi_{861}(34,\cdot)\) \(\chi_{861}(76,\cdot)\) \(\chi_{861}(97,\cdot)\) \(\chi_{861}(181,\cdot)\) \(\chi_{861}(265,\cdot)\) \(\chi_{861}(391,\cdot)\) \(\chi_{861}(475,\cdot)\) \(\chi_{861}(559,\cdot)\) \(\chi_{861}(580,\cdot)\) \(\chi_{861}(622,\cdot)\) \(\chi_{861}(643,\cdot)\) \(\chi_{861}(685,\cdot)\) \(\chi_{861}(727,\cdot)\) \(\chi_{861}(790,\cdot)\) \(\chi_{861}(832,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.40.63172957949423116502957480067191906200305068755882825968063357506461803384975161.1 |
Values on generators
\((575,493,211)\) → \((1,-1,e\left(\frac{21}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 861 }(76, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)