from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1584, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,15,25,9]))
pari: [g,chi] = znchar(Mod(41,1584))
Basic properties
Modulus: | \(1584\) | |
Conductor: | \(792\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{792}(437,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1584.df
\(\chi_{1584}(41,\cdot)\) \(\chi_{1584}(281,\cdot)\) \(\chi_{1584}(425,\cdot)\) \(\chi_{1584}(569,\cdot)\) \(\chi_{1584}(761,\cdot)\) \(\chi_{1584}(1289,\cdot)\) \(\chi_{1584}(1337,\cdot)\) \(\chi_{1584}(1481,\cdot)\)
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: | 30.30.1362726704303137911258873132661821647276632612392121121265156096.1 |
Values on generators
\((991,1189,353,145)\) → \((1,-1,e\left(\frac{5}{6}\right),e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 1584 }(41, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)