from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1225, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,16]))
pari: [g,chi] = znchar(Mod(74,1225))
Basic properties
| Modulus: | \(1225\) | |
| Conductor: | \(245\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{245}(74,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1225.be
\(\chi_{1225}(74,\cdot)\) \(\chi_{1225}(149,\cdot)\) \(\chi_{1225}(249,\cdot)\) \(\chi_{1225}(424,\cdot)\) \(\chi_{1225}(499,\cdot)\) \(\chi_{1225}(599,\cdot)\) \(\chi_{1225}(674,\cdot)\) \(\chi_{1225}(774,\cdot)\) \(\chi_{1225}(849,\cdot)\) \(\chi_{1225}(1024,\cdot)\) \(\chi_{1225}(1124,\cdot)\) \(\chi_{1225}(1199,\cdot)\)
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: | 42.42.8050468075656610214837511220114705524038488445061950919170859146595001220703125.1 |
Values on generators
\((1177,101)\) → \((-1,e\left(\frac{8}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 1225 }(74, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{21}\right)\) |
sage: chi.jacobi_sum(n)