from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4830, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,11,42]))
pari: [g,chi] = znchar(Mod(1991,4830))
Basic properties
Modulus: | \(4830\) | |
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 \(\chi_{483}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4830.da
\(\chi_{4830}(101,\cdot)\) \(\chi_{4830}(131,\cdot)\) \(\chi_{4830}(311,\cdot)\) \(\chi_{4830}(731,\cdot)\) \(\chi_{4830}(761,\cdot)\) \(\chi_{4830}(1181,\cdot)\) \(\chi_{4830}(1361,\cdot)\) \(\chi_{4830}(1991,\cdot)\) \(\chi_{4830}(2201,\cdot)\) \(\chi_{4830}(2441,\cdot)\) \(\chi_{4830}(2651,\cdot)\) \(\chi_{4830}(2831,\cdot)\) \(\chi_{4830}(2861,\cdot)\) \(\chi_{4830}(3071,\cdot)\) \(\chi_{4830}(3251,\cdot)\) \(\chi_{4830}(3491,\cdot)\) \(\chi_{4830}(4121,\cdot)\) \(\chi_{4830}(4511,\cdot)\) \(\chi_{4830}(4721,\cdot)\) \(\chi_{4830}(4751,\cdot)\)
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
\((3221,967,2761,1891)\) → \((-1,1,e\left(\frac{1}{6}\right),e\left(\frac{7}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 4830 }(1991, a) \) | \(1\) | \(1\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)