from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3204, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,0,41]))
pari: [g,chi] = znchar(Mod(199,3204))
Basic properties
| Modulus: | \(3204\) | |
| Conductor: | \(356\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{356}(199,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3204.bs
\(\chi_{3204}(199,\cdot)\) \(\chi_{3204}(307,\cdot)\) \(\chi_{3204}(487,\cdot)\) \(\chi_{3204}(703,\cdot)\) \(\chi_{3204}(811,\cdot)\) \(\chi_{3204}(1063,\cdot)\) \(\chi_{3204}(1315,\cdot)\) \(\chi_{3204}(1495,\cdot)\) \(\chi_{3204}(1531,\cdot)\) \(\chi_{3204}(1711,\cdot)\) \(\chi_{3204}(1963,\cdot)\) \(\chi_{3204}(2215,\cdot)\) \(\chi_{3204}(2323,\cdot)\) \(\chi_{3204}(2539,\cdot)\) \(\chi_{3204}(2719,\cdot)\) \(\chi_{3204}(2827,\cdot)\) \(\chi_{3204}(3043,\cdot)\) \(\chi_{3204}(3079,\cdot)\) \(\chi_{3204}(3151,\cdot)\) \(\chi_{3204}(3187,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | 44.0.11724385642028745774656376755923007752612263949364698259094951647151017819768316744951538808520704.1 |
Values on generators
\((1603,713,181)\) → \((-1,1,e\left(\frac{41}{44}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 3204 }(199, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) |
sage: chi.jacobi_sum(n)