from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2304, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,9,0]))
pari: [g,chi] = znchar(Mod(217,2304))
Basic properties
| Modulus: | \(2304\) | |
| 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 \(\chi_{128}(101,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2304.bl
\(\chi_{2304}(73,\cdot)\) \(\chi_{2304}(217,\cdot)\) \(\chi_{2304}(361,\cdot)\) \(\chi_{2304}(505,\cdot)\) \(\chi_{2304}(649,\cdot)\) \(\chi_{2304}(793,\cdot)\) \(\chi_{2304}(937,\cdot)\) \(\chi_{2304}(1081,\cdot)\) \(\chi_{2304}(1225,\cdot)\) \(\chi_{2304}(1369,\cdot)\) \(\chi_{2304}(1513,\cdot)\) \(\chi_{2304}(1657,\cdot)\) \(\chi_{2304}(1801,\cdot)\) \(\chi_{2304}(1945,\cdot)\) \(\chi_{2304}(2089,\cdot)\) \(\chi_{2304}(2233,\cdot)\)
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
\((1279,2053,1793)\) → \((1,e\left(\frac{9}{32}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 2304 }(217, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)