from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3864, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,33,44,57]))
pari: [g,chi] = znchar(Mod(53,3864))
Basic properties
| Modulus: | \(3864\) | |
| Conductor: | \(3864\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3864.ea
\(\chi_{3864}(53,\cdot)\) \(\chi_{3864}(149,\cdot)\) \(\chi_{3864}(221,\cdot)\) \(\chi_{3864}(389,\cdot)\) \(\chi_{3864}(557,\cdot)\) \(\chi_{3864}(893,\cdot)\) \(\chi_{3864}(1157,\cdot)\) \(\chi_{3864}(1229,\cdot)\) \(\chi_{3864}(1325,\cdot)\) \(\chi_{3864}(1397,\cdot)\) \(\chi_{3864}(1493,\cdot)\) \(\chi_{3864}(1661,\cdot)\) \(\chi_{3864}(1901,\cdot)\) \(\chi_{3864}(1997,\cdot)\) \(\chi_{3864}(2333,\cdot)\) \(\chi_{3864}(2501,\cdot)\) \(\chi_{3864}(2573,\cdot)\) \(\chi_{3864}(2909,\cdot)\) \(\chi_{3864}(3005,\cdot)\) \(\chi_{3864}(3677,\cdot)\)
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
\((967,1933,1289,2761,2857)\) → \((1,-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{19}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3864 }(53, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) |
sage: chi.jacobi_sum(n)