from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(328, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,0,39]))
pari: [g,chi] = znchar(Mod(7,328))
Basic properties
Modulus: | \(328\) | |
Conductor: | \(164\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{164}(7,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 328.bc
\(\chi_{328}(7,\cdot)\) \(\chi_{328}(15,\cdot)\) \(\chi_{328}(47,\cdot)\) \(\chi_{328}(63,\cdot)\) \(\chi_{328}(71,\cdot)\) \(\chi_{328}(95,\cdot)\) \(\chi_{328}(111,\cdot)\) \(\chi_{328}(135,\cdot)\) \(\chi_{328}(151,\cdot)\) \(\chi_{328}(175,\cdot)\) \(\chi_{328}(183,\cdot)\) \(\chi_{328}(199,\cdot)\) \(\chi_{328}(231,\cdot)\) \(\chi_{328}(239,\cdot)\) \(\chi_{328}(263,\cdot)\) \(\chi_{328}(311,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | \(\Q(\zeta_{164})^+\) |
Values on generators
\((247,165,129)\) → \((-1,1,e\left(\frac{39}{40}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 328 }(7, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(i\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)