from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9216, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,7,0]))
pari: [g,chi] = znchar(Mod(289,9216))
Basic properties
| Modulus: | \(9216\) | |
| Conductor: | \(128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{128}(45,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9216.bl
\(\chi_{9216}(289,\cdot)\) \(\chi_{9216}(865,\cdot)\) \(\chi_{9216}(1441,\cdot)\) \(\chi_{9216}(2017,\cdot)\) \(\chi_{9216}(2593,\cdot)\) \(\chi_{9216}(3169,\cdot)\) \(\chi_{9216}(3745,\cdot)\) \(\chi_{9216}(4321,\cdot)\) \(\chi_{9216}(4897,\cdot)\) \(\chi_{9216}(5473,\cdot)\) \(\chi_{9216}(6049,\cdot)\) \(\chi_{9216}(6625,\cdot)\) \(\chi_{9216}(7201,\cdot)\) \(\chi_{9216}(7777,\cdot)\) \(\chi_{9216}(8353,\cdot)\) \(\chi_{9216}(8929,\cdot)\)
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: | \(\Q(\zeta_{128})^+\) |
Values on generators
\((8191,2053,4097)\) → \((1,e\left(\frac{7}{32}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 9216 }(289, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)