from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,9,20]))
pari: [g,chi] = znchar(Mod(58,525))
Basic properties
Modulus: | \(525\) | |
Conductor: | \(175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{175}(58,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 525.bu
\(\chi_{525}(37,\cdot)\) \(\chi_{525}(58,\cdot)\) \(\chi_{525}(67,\cdot)\) \(\chi_{525}(88,\cdot)\) \(\chi_{525}(142,\cdot)\) \(\chi_{525}(163,\cdot)\) \(\chi_{525}(172,\cdot)\) \(\chi_{525}(247,\cdot)\) \(\chi_{525}(277,\cdot)\) \(\chi_{525}(298,\cdot)\) \(\chi_{525}(352,\cdot)\) \(\chi_{525}(373,\cdot)\) \(\chi_{525}(403,\cdot)\) \(\chi_{525}(478,\cdot)\) \(\chi_{525}(487,\cdot)\) \(\chi_{525}(508,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((176,127,451)\) → \((1,e\left(\frac{3}{20}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
\( \chi_{ 525 }(58, a) \) | \(-1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) |
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)