from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1600, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,15,28]))
pari: [g,chi] = znchar(Mod(9,1600))
Basic properties
Modulus: | \(1600\) | |
Conductor: | \(800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{800}(509,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1600.cg
\(\chi_{1600}(9,\cdot)\) \(\chi_{1600}(89,\cdot)\) \(\chi_{1600}(169,\cdot)\) \(\chi_{1600}(329,\cdot)\) \(\chi_{1600}(409,\cdot)\) \(\chi_{1600}(489,\cdot)\) \(\chi_{1600}(569,\cdot)\) \(\chi_{1600}(729,\cdot)\) \(\chi_{1600}(809,\cdot)\) \(\chi_{1600}(889,\cdot)\) \(\chi_{1600}(969,\cdot)\) \(\chi_{1600}(1129,\cdot)\) \(\chi_{1600}(1209,\cdot)\) \(\chi_{1600}(1289,\cdot)\) \(\chi_{1600}(1369,\cdot)\) \(\chi_{1600}(1529,\cdot)\)
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: | 40.40.15474250491067253436239052800000000000000000000000000000000000000000000000000000000000000000000.1 |
Values on generators
\((1151,901,577)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{7}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 1600 }(9, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{40}\right)\) | \(i\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) |
sage: chi.jacobi_sum(n)