from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3234, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,29,21]))
pari: [g,chi] = znchar(Mod(2749,3234))
Basic properties
Modulus: | \(3234\) | |
Conductor: | \(539\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{539}(54,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3234.br
\(\chi_{3234}(241,\cdot)\) \(\chi_{3234}(439,\cdot)\) \(\chi_{3234}(703,\cdot)\) \(\chi_{3234}(1165,\cdot)\) \(\chi_{3234}(1363,\cdot)\) \(\chi_{3234}(1627,\cdot)\) \(\chi_{3234}(1825,\cdot)\) \(\chi_{3234}(2287,\cdot)\) \(\chi_{3234}(2551,\cdot)\) \(\chi_{3234}(2749,\cdot)\) \(\chi_{3234}(3013,\cdot)\) \(\chi_{3234}(3211,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((1079,199,2059)\) → \((1,e\left(\frac{29}{42}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3234 }(2749, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) |
sage: chi.jacobi_sum(n)