from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2233, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,18,45]))
pari: [g,chi] = znchar(Mod(800,2233))
Basic properties
Modulus: | \(2233\) | |
Conductor: | \(2233\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | 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 2233.cn
\(\chi_{2233}(46,\cdot)\) \(\chi_{2233}(128,\cdot)\) \(\chi_{2233}(249,\cdot)\) \(\chi_{2233}(534,\cdot)\) \(\chi_{2233}(655,\cdot)\) \(\chi_{2233}(800,\cdot)\) \(\chi_{2233}(998,\cdot)\) \(\chi_{2233}(1003,\cdot)\) \(\chi_{2233}(1201,\cdot)\) \(\chi_{2233}(1206,\cdot)\) \(\chi_{2233}(1404,\cdot)\) \(\chi_{2233}(1612,\cdot)\) \(\chi_{2233}(1810,\cdot)\) \(\chi_{2233}(1955,\cdot)\) \(\chi_{2233}(2076,\cdot)\) \(\chi_{2233}(2158,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((1277,1828,2003)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{3}{10}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
\( \chi_{ 2233 }(800, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{4}{5}\right)\) |
sage: chi.jacobi_sum(n)