sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(53, base_ring=CyclotomicField(26))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([24]))
pari: [g,chi] = znchar(Mod(10,53))
Basic properties
| Modulus: | \(53\) | |
| Conductor: | \(53\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(13\) | 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 53.d
\(\chi_{53}(10,\cdot)\) \(\chi_{53}(13,\cdot)\) \(\chi_{53}(15,\cdot)\) \(\chi_{53}(16,\cdot)\) \(\chi_{53}(24,\cdot)\) \(\chi_{53}(28,\cdot)\) \(\chi_{53}(36,\cdot)\) \(\chi_{53}(42,\cdot)\) \(\chi_{53}(44,\cdot)\) \(\chi_{53}(46,\cdot)\) \(\chi_{53}(47,\cdot)\) \(\chi_{53}(49,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{13})\) |
| Fixed field: | 13.13.491258904256726154641.1 |
Values on generators
\(2\) → \(e\left(\frac{12}{13}\right)\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \(1\) | \(1\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{53}(10,\cdot)) = \sum_{r\in \Z/53\Z} \chi_{53}(10,r) e\left(\frac{2r}{53}\right) = 4.4724868635+-5.7442894474i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{53}(10,\cdot),\chi_{53}(1,\cdot)) = \sum_{r\in \Z/53\Z} \chi_{53}(10,r) \chi_{53}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{53}(10,·))
= \sum_{r \in \Z/53\Z}
\chi_{53}(10,r) e\left(\frac{1 r + 2 r^{-1}}{53}\right)
= 4.4841488958+-1.1052434374i \)