sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(483, base_ring=CyclotomicField(22))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,0,12]))
pari: [g,chi] = znchar(Mod(64,483))
Basic properties
| Modulus: | \(483\) | |
| Conductor: | \(23\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(11\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{23}(18,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 483.q
\(\chi_{483}(64,\cdot)\) \(\chi_{483}(85,\cdot)\) \(\chi_{483}(127,\cdot)\) \(\chi_{483}(169,\cdot)\) \(\chi_{483}(190,\cdot)\) \(\chi_{483}(211,\cdot)\) \(\chi_{483}(232,\cdot)\) \(\chi_{483}(358,\cdot)\) \(\chi_{483}(400,\cdot)\) \(\chi_{483}(463,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | \(\Q(\zeta_{23})^+\) |
Values on generators
\((323,346,442)\) → \((1,1,e\left(\frac{6}{11}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{483}(64,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(64,r) e\left(\frac{2r}{483}\right) = 4.7300087956+0.7918439198i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{483}(64,\cdot),\chi_{483}(1,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(64,r) \chi_{483}(1,1-r) = -5 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{483}(64,·))
= \sum_{r \in \Z/483\Z}
\chi_{483}(64,r) e\left(\frac{1 r + 2 r^{-1}}{483}\right)
= 16.7998558346+4.9328827506i \)