from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9216, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,25,16]))
pari: [g,chi] = znchar(Mod(287,9216))
Basic properties
| Modulus: | \(9216\) | |
| 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 \(\chi_{384}(347,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9216.bm
\(\chi_{9216}(287,\cdot)\) \(\chi_{9216}(863,\cdot)\) \(\chi_{9216}(1439,\cdot)\) \(\chi_{9216}(2015,\cdot)\) \(\chi_{9216}(2591,\cdot)\) \(\chi_{9216}(3167,\cdot)\) \(\chi_{9216}(3743,\cdot)\) \(\chi_{9216}(4319,\cdot)\) \(\chi_{9216}(4895,\cdot)\) \(\chi_{9216}(5471,\cdot)\) \(\chi_{9216}(6047,\cdot)\) \(\chi_{9216}(6623,\cdot)\) \(\chi_{9216}(7199,\cdot)\) \(\chi_{9216}(7775,\cdot)\) \(\chi_{9216}(8351,\cdot)\) \(\chi_{9216}(8927,\cdot)\)
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
\((8191,2053,4097)\) → \((-1,e\left(\frac{25}{32}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 9216 }(287, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)