from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8280, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,55,0,24]))
pari: [g,chi] = znchar(Mod(131,8280))
Basic properties
| Modulus: | \(8280\) | |
| Conductor: | \(1656\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1656}(131,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8280.fs
\(\chi_{8280}(131,\cdot)\) \(\chi_{8280}(371,\cdot)\) \(\chi_{8280}(491,\cdot)\) \(\chi_{8280}(731,\cdot)\) \(\chi_{8280}(1451,\cdot)\) \(\chi_{8280}(2171,\cdot)\) \(\chi_{8280}(2651,\cdot)\) \(\chi_{8280}(2891,\cdot)\) \(\chi_{8280}(3251,\cdot)\) \(\chi_{8280}(3371,\cdot)\) \(\chi_{8280}(4451,\cdot)\) \(\chi_{8280}(4811,\cdot)\) \(\chi_{8280}(5411,\cdot)\) \(\chi_{8280}(5891,\cdot)\) \(\chi_{8280}(6131,\cdot)\) \(\chi_{8280}(6251,\cdot)\) \(\chi_{8280}(6971,\cdot)\) \(\chi_{8280}(7211,\cdot)\) \(\chi_{8280}(7571,\cdot)\) \(\chi_{8280}(7691,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((2071,4141,4601,1657,3961)\) → \((-1,-1,e\left(\frac{5}{6}\right),1,e\left(\frac{4}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8280 }(131, a) \) | \(1\) | \(1\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) |
sage: chi.jacobi_sum(n)