from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6045, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,10,5,16]))
pari: [g,chi] = znchar(Mod(2699,6045))
Basic properties
Modulus: | \(6045\) | |
Conductor: | \(6045\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | 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 6045.jb
\(\chi_{6045}(2699,\cdot)\) \(\chi_{6045}(3164,\cdot)\) \(\chi_{6045}(3674,\cdot)\) \(\chi_{6045}(4139,\cdot)\) \(\chi_{6045}(4844,\cdot)\) \(\chi_{6045}(5309,\cdot)\) \(\chi_{6045}(5429,\cdot)\) \(\chi_{6045}(5894,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((4031,4837,1861,2731)\) → \((-1,-1,i,e\left(\frac{4}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
\( \chi_{ 6045 }(2699, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
sage: chi.jacobi_sum(n)