from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1320, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,0,10,15,16]))
pari: [g,chi] = znchar(Mod(113,1320))
Basic properties
| Modulus: | \(1320\) | |
| 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 \(\chi_{165}(113,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1320.dl
\(\chi_{1320}(113,\cdot)\) \(\chi_{1320}(137,\cdot)\) \(\chi_{1320}(257,\cdot)\) \(\chi_{1320}(377,\cdot)\) \(\chi_{1320}(713,\cdot)\) \(\chi_{1320}(977,\cdot)\) \(\chi_{1320}(1193,\cdot)\) \(\chi_{1320}(1313,\cdot)\)
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
\((991,661,881,1057,1201)\) → \((1,1,-1,-i,e\left(\frac{4}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1320 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(-i\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)