from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1305, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([28,21,27]))
pari: [g,chi] = znchar(Mod(709,1305))
Basic properties
| Modulus: | \(1305\) | |
| Conductor: | \(1305\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | 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 1305.co
\(\chi_{1305}(4,\cdot)\) \(\chi_{1305}(34,\cdot)\) \(\chi_{1305}(274,\cdot)\) \(\chi_{1305}(439,\cdot)\) \(\chi_{1305}(499,\cdot)\) \(\chi_{1305}(544,\cdot)\) \(\chi_{1305}(589,\cdot)\) \(\chi_{1305}(709,\cdot)\) \(\chi_{1305}(904,\cdot)\) \(\chi_{1305}(934,\cdot)\) \(\chi_{1305}(979,\cdot)\) \(\chi_{1305}(1024,\cdot)\)
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
\((146,262,901)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{9}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1305 }(709, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(1\) | \(e\left(\frac{11}{14}\right)\) |
sage: chi.jacobi_sum(n)