from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1920, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,9,16,0]))
pari: [g,chi] = znchar(Mod(101,1920))
Basic properties
| Modulus: | \(1920\) | |
| 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 \(\chi_{384}(101,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1920.de
\(\chi_{1920}(101,\cdot)\) \(\chi_{1920}(221,\cdot)\) \(\chi_{1920}(341,\cdot)\) \(\chi_{1920}(461,\cdot)\) \(\chi_{1920}(581,\cdot)\) \(\chi_{1920}(701,\cdot)\) \(\chi_{1920}(821,\cdot)\) \(\chi_{1920}(941,\cdot)\) \(\chi_{1920}(1061,\cdot)\) \(\chi_{1920}(1181,\cdot)\) \(\chi_{1920}(1301,\cdot)\) \(\chi_{1920}(1421,\cdot)\) \(\chi_{1920}(1541,\cdot)\) \(\chi_{1920}(1661,\cdot)\) \(\chi_{1920}(1781,\cdot)\) \(\chi_{1920}(1901,\cdot)\)
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: | 32.0.135104323545903136978453058557785670637514001130337144105502507008.1 |
Values on generators
\((511,901,641,1537)\) → \((1,e\left(\frac{9}{32}\right),-1,1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1920 }(101, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(i\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) |
sage: chi.jacobi_sum(n)