from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3456, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,13,16]))
pari: [g,chi] = znchar(Mod(107,3456))
Basic properties
| Modulus: | \(3456\) | |
| 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 \(\chi_{384}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3456.bu
\(\chi_{3456}(107,\cdot)\) \(\chi_{3456}(323,\cdot)\) \(\chi_{3456}(539,\cdot)\) \(\chi_{3456}(755,\cdot)\) \(\chi_{3456}(971,\cdot)\) \(\chi_{3456}(1187,\cdot)\) \(\chi_{3456}(1403,\cdot)\) \(\chi_{3456}(1619,\cdot)\) \(\chi_{3456}(1835,\cdot)\) \(\chi_{3456}(2051,\cdot)\) \(\chi_{3456}(2267,\cdot)\) \(\chi_{3456}(2483,\cdot)\) \(\chi_{3456}(2699,\cdot)\) \(\chi_{3456}(2915,\cdot)\) \(\chi_{3456}(3131,\cdot)\) \(\chi_{3456}(3347,\cdot)\)
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
\((2431,2053,2945)\) → \((-1,e\left(\frac{13}{32}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 3456 }(107, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)