from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1815, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,5,14]))
pari: [g,chi] = znchar(Mod(1492,1815))
Basic properties
Modulus: | \(1815\) | |
Conductor: | \(55\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{55}(7,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1815.x
\(\chi_{1815}(112,\cdot)\) \(\chi_{1815}(118,\cdot)\) \(\chi_{1815}(403,\cdot)\) \(\chi_{1815}(457,\cdot)\) \(\chi_{1815}(838,\cdot)\) \(\chi_{1815}(1183,\cdot)\) \(\chi_{1815}(1207,\cdot)\) \(\chi_{1815}(1492,\cdot)\)
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: | \(\Q(\zeta_{55})^+\) |
Values on generators
\((1211,727,1696)\) → \((1,i,e\left(\frac{7}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
\( \chi_{ 1815 }(1492, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)