from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9600, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,29,0,24]))
pari: [g,chi] = znchar(Mod(43,9600))
Basic properties
Modulus: | \(9600\) | |
Conductor: | \(640\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{640}(43,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9600.ew
\(\chi_{9600}(43,\cdot)\) \(\chi_{9600}(307,\cdot)\) \(\chi_{9600}(1243,\cdot)\) \(\chi_{9600}(1507,\cdot)\) \(\chi_{9600}(2443,\cdot)\) \(\chi_{9600}(2707,\cdot)\) \(\chi_{9600}(3643,\cdot)\) \(\chi_{9600}(3907,\cdot)\) \(\chi_{9600}(4843,\cdot)\) \(\chi_{9600}(5107,\cdot)\) \(\chi_{9600}(6043,\cdot)\) \(\chi_{9600}(6307,\cdot)\) \(\chi_{9600}(7243,\cdot)\) \(\chi_{9600}(7507,\cdot)\) \(\chi_{9600}(8443,\cdot)\) \(\chi_{9600}(8707,\cdot)\)
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: | 32.32.187072209578355573530071658587684226515959365500928000000000000000000000000.1 |
Values on generators
\((4351,901,6401,5377)\) → \((-1,e\left(\frac{29}{32}\right),1,-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 9600 }(43, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(-i\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) |
sage: chi.jacobi_sum(n)