from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(576, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,39,40]))
pari: [g,chi] = znchar(Mod(149,576))
Basic properties
Modulus: | \(576\) | |
Conductor: | \(576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 576.bn
\(\chi_{576}(5,\cdot)\) \(\chi_{576}(29,\cdot)\) \(\chi_{576}(77,\cdot)\) \(\chi_{576}(101,\cdot)\) \(\chi_{576}(149,\cdot)\) \(\chi_{576}(173,\cdot)\) \(\chi_{576}(221,\cdot)\) \(\chi_{576}(245,\cdot)\) \(\chi_{576}(293,\cdot)\) \(\chi_{576}(317,\cdot)\) \(\chi_{576}(365,\cdot)\) \(\chi_{576}(389,\cdot)\) \(\chi_{576}(437,\cdot)\) \(\chi_{576}(461,\cdot)\) \(\chi_{576}(509,\cdot)\) \(\chi_{576}(533,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((127,325,65)\) → \((1,e\left(\frac{13}{16}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 576 }(149, a) \) | \(-1\) | \(1\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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)