from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(372, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,0,4]))
pari: [g,chi] = znchar(Mod(19,372))
Basic properties
| Modulus: | \(372\) | |
| Conductor: | \(124\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{124}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 372.z
\(\chi_{372}(7,\cdot)\) \(\chi_{372}(19,\cdot)\) \(\chi_{372}(103,\cdot)\) \(\chi_{372}(175,\cdot)\) \(\chi_{372}(235,\cdot)\) \(\chi_{372}(307,\cdot)\) \(\chi_{372}(319,\cdot)\) \(\chi_{372}(355,\cdot)\)
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: | 30.0.615215540441622698713738389172402189599059721846784.1 |
Values on generators
\((187,125,313)\) → \((-1,1,e\left(\frac{2}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(35\) |
| \( \chi_{ 372 }(19, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) |
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)