from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1395, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([10,45,52]))
pari: [g,chi] = znchar(Mod(173,1395))
Basic properties
Modulus: | \(1395\) | |
Conductor: | \(1395\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1395.ek
\(\chi_{1395}(173,\cdot)\) \(\chi_{1395}(227,\cdot)\) \(\chi_{1395}(293,\cdot)\) \(\chi_{1395}(338,\cdot)\) \(\chi_{1395}(443,\cdot)\) \(\chi_{1395}(452,\cdot)\) \(\chi_{1395}(572,\cdot)\) \(\chi_{1395}(578,\cdot)\) \(\chi_{1395}(608,\cdot)\) \(\chi_{1395}(617,\cdot)\) \(\chi_{1395}(722,\cdot)\) \(\chi_{1395}(857,\cdot)\) \(\chi_{1395}(887,\cdot)\) \(\chi_{1395}(968,\cdot)\) \(\chi_{1395}(1247,\cdot)\) \(\chi_{1395}(1343,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((776,1117,406)\) → \((e\left(\frac{1}{6}\right),-i,e\left(\frac{13}{15}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 1395 }(173, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) |
sage: chi.jacobi_sum(n)