from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1176, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,0,29]))
pari: [g,chi] = znchar(Mod(103,1176))
Basic properties
| Modulus: | \(1176\) | |
| Conductor: | \(196\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{196}(103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1176.bz
\(\chi_{1176}(103,\cdot)\) \(\chi_{1176}(199,\cdot)\) \(\chi_{1176}(271,\cdot)\) \(\chi_{1176}(367,\cdot)\) \(\chi_{1176}(439,\cdot)\) \(\chi_{1176}(535,\cdot)\) \(\chi_{1176}(703,\cdot)\) \(\chi_{1176}(775,\cdot)\) \(\chi_{1176}(871,\cdot)\) \(\chi_{1176}(943,\cdot)\) \(\chi_{1176}(1039,\cdot)\) \(\chi_{1176}(1111,\cdot)\)
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: | \(\Q(\zeta_{196})^+\) |
Values on generators
\((295,589,785,1081)\) → \((-1,1,1,e\left(\frac{29}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 1176 }(103, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{21}\right)\) |
sage: chi.jacobi_sum(n)