from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(715, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([15,54,10]))
pari: [g,chi] = znchar(Mod(17,715))
Basic properties
Modulus: | \(715\) | |
Conductor: | \(715\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | 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 715.cx
\(\chi_{715}(17,\cdot)\) \(\chi_{715}(62,\cdot)\) \(\chi_{715}(127,\cdot)\) \(\chi_{715}(173,\cdot)\) \(\chi_{715}(238,\cdot)\) \(\chi_{715}(277,\cdot)\) \(\chi_{715}(283,\cdot)\) \(\chi_{715}(303,\cdot)\) \(\chi_{715}(348,\cdot)\) \(\chi_{715}(387,\cdot)\) \(\chi_{715}(413,\cdot)\) \(\chi_{715}(563,\cdot)\) \(\chi_{715}(602,\cdot)\) \(\chi_{715}(667,\cdot)\) \(\chi_{715}(673,\cdot)\) \(\chi_{715}(712,\cdot)\)
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
\((287,651,496)\) → \((i,e\left(\frac{9}{10}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(14\) | \(16\) |
\( \chi_{ 715 }(17, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(i\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) |
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)