from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5550, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,9,10]))
pari: [g,chi] = znchar(Mod(887,5550))
Basic properties
| Modulus: | \(5550\) | |
| Conductor: | \(2775\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2775}(887,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5550.cm
\(\chi_{5550}(887,\cdot)\) \(\chi_{5550}(1553,\cdot)\) \(\chi_{5550}(1997,\cdot)\) \(\chi_{5550}(2663,\cdot)\) \(\chi_{5550}(3773,\cdot)\) \(\chi_{5550}(4217,\cdot)\) \(\chi_{5550}(4883,\cdot)\) \(\chi_{5550}(5327,\cdot)\)
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
\((3701,1777,2851)\) → \((-1,e\left(\frac{9}{20}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
| \( \chi_{ 5550 }(887, a) \) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)