from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2548, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,11,28]))
pari: [g,chi] = znchar(Mod(61,2548))
Basic properties
| Modulus: | \(2548\) | |
| Conductor: | \(637\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{637}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2548.dw
\(\chi_{2548}(61,\cdot)\) \(\chi_{2548}(185,\cdot)\) \(\chi_{2548}(425,\cdot)\) \(\chi_{2548}(549,\cdot)\) \(\chi_{2548}(789,\cdot)\) \(\chi_{2548}(1153,\cdot)\) \(\chi_{2548}(1277,\cdot)\) \(\chi_{2548}(1517,\cdot)\) \(\chi_{2548}(1641,\cdot)\) \(\chi_{2548}(2005,\cdot)\) \(\chi_{2548}(2245,\cdot)\) \(\chi_{2548}(2369,\cdot)\)
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
\((1275,885,197)\) → \((1,e\left(\frac{11}{42}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 2548 }(61, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(-1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) |
sage: chi.jacobi_sum(n)