from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1980, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,10,5,4]))
pari: [g,chi] = znchar(Mod(917,1980))
Basic properties
Modulus: | \(1980\) | |
Conductor: | \(165\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{165}(92,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1980.cr
\(\chi_{1980}(53,\cdot)\) \(\chi_{1980}(377,\cdot)\) \(\chi_{1980}(773,\cdot)\) \(\chi_{1980}(917,\cdot)\) \(\chi_{1980}(1313,\cdot)\) \(\chi_{1980}(1457,\cdot)\) \(\chi_{1980}(1637,\cdot)\) \(\chi_{1980}(1853,\cdot)\)
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
\((991,1541,397,541)\) → \((1,-1,i,e\left(\frac{1}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 1980 }(917, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(i\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)