from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(400, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,7]))
pari: [g,chi] = znchar(Mod(303,400))
Basic properties
| Modulus: | \(400\) | |
| Conductor: | \(100\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{100}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 400.bi
\(\chi_{400}(47,\cdot)\) \(\chi_{400}(63,\cdot)\) \(\chi_{400}(127,\cdot)\) \(\chi_{400}(223,\cdot)\) \(\chi_{400}(287,\cdot)\) \(\chi_{400}(303,\cdot)\) \(\chi_{400}(367,\cdot)\) \(\chi_{400}(383,\cdot)\)
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: | \(\Q(\zeta_{100})^+\) |
Values on generators
\((351,101,177)\) → \((-1,1,e\left(\frac{7}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 400 }(303, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(i\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) |
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)