from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(900, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,20,24]))
pari: [g,chi] = znchar(Mod(61,900))
Basic properties
| Modulus: | \(900\) | |
| Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{225}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 900.bg
\(\chi_{900}(61,\cdot)\) \(\chi_{900}(121,\cdot)\) \(\chi_{900}(241,\cdot)\) \(\chi_{900}(421,\cdot)\) \(\chi_{900}(481,\cdot)\) \(\chi_{900}(661,\cdot)\) \(\chi_{900}(781,\cdot)\) \(\chi_{900}(841,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | Number field defined by a degree 15 polynomial |
Values on generators
\((451,101,577)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{4}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 900 }(61, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) |
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)