from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10000, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,5,1]))
pari: [g,chi] = znchar(Mod(1307,10000))
Basic properties
Modulus: | \(10000\) | |
Conductor: | \(400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{400}(27,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 10000.bd
\(\chi_{10000}(1307,\cdot)\) \(\chi_{10000}(1443,\cdot)\) \(\chi_{10000}(3443,\cdot)\) \(\chi_{10000}(5307,\cdot)\) \(\chi_{10000}(7307,\cdot)\) \(\chi_{10000}(7443,\cdot)\) \(\chi_{10000}(9307,\cdot)\) \(\chi_{10000}(9443,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | 20.20.104857600000000000000000000000000000000000.2 |
Values on generators
\((8751,2501,9377)\) → \((-1,i,e\left(\frac{1}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 10000 }(1307, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(i\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) |
sage: chi.jacobi_sum(n)