from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3712, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,5,16]))
pari: [g,chi] = znchar(Mod(1333,3712))
Basic properties
| Modulus: | \(3712\) | |
| Conductor: | \(3712\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | 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 3712.cg
\(\chi_{3712}(173,\cdot)\) \(\chi_{3712}(405,\cdot)\) \(\chi_{3712}(637,\cdot)\) \(\chi_{3712}(869,\cdot)\) \(\chi_{3712}(1101,\cdot)\) \(\chi_{3712}(1333,\cdot)\) \(\chi_{3712}(1565,\cdot)\) \(\chi_{3712}(1797,\cdot)\) \(\chi_{3712}(2029,\cdot)\) \(\chi_{3712}(2261,\cdot)\) \(\chi_{3712}(2493,\cdot)\) \(\chi_{3712}(2725,\cdot)\) \(\chi_{3712}(2957,\cdot)\) \(\chi_{3712}(3189,\cdot)\) \(\chi_{3712}(3421,\cdot)\) \(\chi_{3712}(3653,\cdot)\)
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: | 32.32.785411287106652838033363851140085365286337539393649349536153620008513161719185408.1 |
Values on generators
\((639,2437,2177)\) → \((1,e\left(\frac{5}{32}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 3712 }(1333, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) |
sage: chi.jacobi_sum(n)