from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(564, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([0,0,30]))
pari: [g,chi] = znchar(Mod(553,564))
Basic properties
Modulus: | \(564\) | |
Conductor: | \(47\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(23\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{47}(36,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 564.i
\(\chi_{564}(25,\cdot)\) \(\chi_{564}(37,\cdot)\) \(\chi_{564}(49,\cdot)\) \(\chi_{564}(61,\cdot)\) \(\chi_{564}(97,\cdot)\) \(\chi_{564}(121,\cdot)\) \(\chi_{564}(145,\cdot)\) \(\chi_{564}(157,\cdot)\) \(\chi_{564}(169,\cdot)\) \(\chi_{564}(205,\cdot)\) \(\chi_{564}(241,\cdot)\) \(\chi_{564}(253,\cdot)\) \(\chi_{564}(277,\cdot)\) \(\chi_{564}(289,\cdot)\) \(\chi_{564}(337,\cdot)\) \(\chi_{564}(361,\cdot)\) \(\chi_{564}(385,\cdot)\) \(\chi_{564}(397,\cdot)\) \(\chi_{564}(457,\cdot)\) \(\chi_{564}(529,\cdot)\) \(\chi_{564}(541,\cdot)\) \(\chi_{564}(553,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
Fixed field: | Number field defined by a degree 23 polynomial |
Values on generators
\((283,377,193)\) → \((1,1,e\left(\frac{15}{23}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 564 }(553, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) |
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)