from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9660, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,0,22,45]))
pari: [g,chi] = znchar(Mod(5861,9660))
Basic properties
| Modulus: | \(9660\) | |
| Conductor: | \(483\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{483}(65,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9660.gk
\(\chi_{9660}(221,\cdot)\) \(\chi_{9660}(401,\cdot)\) \(\chi_{9660}(641,\cdot)\) \(\chi_{9660}(1661,\cdot)\) \(\chi_{9660}(1901,\cdot)\) \(\chi_{9660}(2081,\cdot)\) \(\chi_{9660}(2321,\cdot)\) \(\chi_{9660}(2501,\cdot)\) \(\chi_{9660}(3161,\cdot)\) \(\chi_{9660}(4421,\cdot)\) \(\chi_{9660}(4841,\cdot)\) \(\chi_{9660}(5021,\cdot)\) \(\chi_{9660}(5261,\cdot)\) \(\chi_{9660}(5861,\cdot)\) \(\chi_{9660}(7121,\cdot)\) \(\chi_{9660}(7541,\cdot)\) \(\chi_{9660}(7781,\cdot)\) \(\chi_{9660}(8621,\cdot)\) \(\chi_{9660}(8801,\cdot)\) \(\chi_{9660}(9221,\cdot)\)
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
\((4831,3221,5797,2761,6721)\) → \((1,-1,1,e\left(\frac{1}{3}\right),e\left(\frac{15}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
| \( \chi_{ 9660 }(5861, a) \) | \(1\) | \(1\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)