from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(113, base_ring=CyclotomicField(56))
M = H._module
chi = DirichletCharacter(H, M([51]))
pari: [g,chi] = znchar(Mod(61,113))
Basic properties
Modulus: | \(113\) | |
Conductor: | \(113\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(56\) | 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 113.i
\(\chi_{113}(9,\cdot)\) \(\chi_{113}(11,\cdot)\) \(\chi_{113}(13,\cdot)\) \(\chi_{113}(22,\cdot)\) \(\chi_{113}(25,\cdot)\) \(\chi_{113}(26,\cdot)\) \(\chi_{113}(31,\cdot)\) \(\chi_{113}(36,\cdot)\) \(\chi_{113}(41,\cdot)\) \(\chi_{113}(50,\cdot)\) \(\chi_{113}(51,\cdot)\) \(\chi_{113}(52,\cdot)\) \(\chi_{113}(61,\cdot)\) \(\chi_{113}(62,\cdot)\) \(\chi_{113}(63,\cdot)\) \(\chi_{113}(72,\cdot)\) \(\chi_{113}(77,\cdot)\) \(\chi_{113}(82,\cdot)\) \(\chi_{113}(87,\cdot)\) \(\chi_{113}(88,\cdot)\) \(\chi_{113}(91,\cdot)\) \(\chi_{113}(100,\cdot)\) \(\chi_{113}(102,\cdot)\) \(\chi_{113}(104,\cdot)\)
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: | $\Q(\zeta_{56})$ |
Fixed field: | Number field defined by a degree 56 polynomial |
Values on generators
\(3\) → \(e\left(\frac{51}{56}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 113 }(61, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) |
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)