from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4608, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,3,16]))
pari: [g,chi] = znchar(Mod(431,4608))
Basic properties
| Modulus: | \(4608\) | |
| Conductor: | \(384\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{384}(131,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4608.bm
\(\chi_{4608}(143,\cdot)\) \(\chi_{4608}(431,\cdot)\) \(\chi_{4608}(719,\cdot)\) \(\chi_{4608}(1007,\cdot)\) \(\chi_{4608}(1295,\cdot)\) \(\chi_{4608}(1583,\cdot)\) \(\chi_{4608}(1871,\cdot)\) \(\chi_{4608}(2159,\cdot)\) \(\chi_{4608}(2447,\cdot)\) \(\chi_{4608}(2735,\cdot)\) \(\chi_{4608}(3023,\cdot)\) \(\chi_{4608}(3311,\cdot)\) \(\chi_{4608}(3599,\cdot)\) \(\chi_{4608}(3887,\cdot)\) \(\chi_{4608}(4175,\cdot)\) \(\chi_{4608}(4463,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | 32.32.135104323545903136978453058557785670637514001130337144105502507008.1 |
Values on generators
\((3583,2053,4097)\) → \((-1,e\left(\frac{3}{32}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 4608 }(431, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)