sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(53, base_ring=CyclotomicField(26))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([19]))
pari: [g,chi] = znchar(Mod(38,53))
Basic properties
| Modulus: | \(53\) | |
| Conductor: | \(53\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | 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.e
\(\chi_{53}(4,\cdot)\) \(\chi_{53}(6,\cdot)\) \(\chi_{53}(7,\cdot)\) \(\chi_{53}(9,\cdot)\) \(\chi_{53}(11,\cdot)\) \(\chi_{53}(17,\cdot)\) \(\chi_{53}(25,\cdot)\) \(\chi_{53}(29,\cdot)\) \(\chi_{53}(37,\cdot)\) \(\chi_{53}(38,\cdot)\) \(\chi_{53}(40,\cdot)\) \(\chi_{53}(43,\cdot)\)
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: | \(\Q(\zeta_{53})^+\) |
Values on generators
\(2\) → \(e\left(\frac{19}{26}\right)\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \(1\) | \(1\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{53}(38,\cdot)) = \sum_{r\in \Z/53\Z} \chi_{53}(38,r) e\left(\frac{2r}{53}\right) = -7.115717931+1.5383622221i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{53}(38,\cdot),\chi_{53}(1,\cdot)) = \sum_{r\in \Z/53\Z} \chi_{53}(38,r) \chi_{53}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{53}(38,·))
= \sum_{r \in \Z/53\Z}
\chi_{53}(38,r) e\left(\frac{1 r + 2 r^{-1}}{53}\right)
= 5.8955450896+-6.6546947402i \)