from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1056, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,5,20,8]))
pari: [g,chi] = znchar(Mod(59,1056))
Basic properties
| Modulus: | \(1056\) | |
| Conductor: | \(1056\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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 1056.ch
\(\chi_{1056}(59,\cdot)\) \(\chi_{1056}(179,\cdot)\) \(\chi_{1056}(203,\cdot)\) \(\chi_{1056}(251,\cdot)\) \(\chi_{1056}(323,\cdot)\) \(\chi_{1056}(443,\cdot)\) \(\chi_{1056}(467,\cdot)\) \(\chi_{1056}(515,\cdot)\) \(\chi_{1056}(587,\cdot)\) \(\chi_{1056}(707,\cdot)\) \(\chi_{1056}(731,\cdot)\) \(\chi_{1056}(779,\cdot)\) \(\chi_{1056}(851,\cdot)\) \(\chi_{1056}(971,\cdot)\) \(\chi_{1056}(995,\cdot)\) \(\chi_{1056}(1043,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((991,133,353,673)\) → \((-1,e\left(\frac{1}{8}\right),-1,e\left(\frac{1}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 1056 }(59, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(-i\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) |
sage: chi.jacobi_sum(n)