from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1664, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,9,16]))
pari: [g,chi] = znchar(Mod(1637,1664))
Basic properties
Modulus: | \(1664\) | |
Conductor: | \(1664\) | 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 1664.cr
\(\chi_{1664}(77,\cdot)\) \(\chi_{1664}(181,\cdot)\) \(\chi_{1664}(285,\cdot)\) \(\chi_{1664}(389,\cdot)\) \(\chi_{1664}(493,\cdot)\) \(\chi_{1664}(597,\cdot)\) \(\chi_{1664}(701,\cdot)\) \(\chi_{1664}(805,\cdot)\) \(\chi_{1664}(909,\cdot)\) \(\chi_{1664}(1013,\cdot)\) \(\chi_{1664}(1117,\cdot)\) \(\chi_{1664}(1221,\cdot)\) \(\chi_{1664}(1325,\cdot)\) \(\chi_{1664}(1429,\cdot)\) \(\chi_{1664}(1533,\cdot)\) \(\chi_{1664}(1637,\cdot)\)
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: | 32.32.2088443876129429457733048543333054873029337200425307489036314630977400864768.1 |
Values on generators
\((1535,261,769)\) → \((1,e\left(\frac{9}{32}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
\( \chi_{ 1664 }(1637, a) \) | \(1\) | \(1\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) |
sage: chi.jacobi_sum(n)