from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(861, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,0,11]))
pari: [g,chi] = znchar(Mod(848,861))
Basic properties
Modulus: | \(861\) | |
Conductor: | \(123\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{123}(110,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 861.ca
\(\chi_{861}(29,\cdot)\) \(\chi_{861}(71,\cdot)\) \(\chi_{861}(134,\cdot)\) \(\chi_{861}(176,\cdot)\) \(\chi_{861}(218,\cdot)\) \(\chi_{861}(239,\cdot)\) \(\chi_{861}(281,\cdot)\) \(\chi_{861}(302,\cdot)\) \(\chi_{861}(386,\cdot)\) \(\chi_{861}(470,\cdot)\) \(\chi_{861}(596,\cdot)\) \(\chi_{861}(680,\cdot)\) \(\chi_{861}(764,\cdot)\) \(\chi_{861}(785,\cdot)\) \(\chi_{861}(827,\cdot)\) \(\chi_{861}(848,\cdot)\)
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: | \(\Q(\zeta_{123})^+\) |
Values on generators
\((575,493,211)\) → \((-1,1,e\left(\frac{11}{40}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 861 }(848, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) |
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)