sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(483, base_ring=CyclotomicField(66))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,44,12]))
pari: [g,chi] = znchar(Mod(4,483))
Basic properties
Modulus: | \(483\) | |
Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{161}(4,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 483.y
\(\chi_{483}(4,\cdot)\) \(\chi_{483}(16,\cdot)\) \(\chi_{483}(25,\cdot)\) \(\chi_{483}(58,\cdot)\) \(\chi_{483}(100,\cdot)\) \(\chi_{483}(121,\cdot)\) \(\chi_{483}(142,\cdot)\) \(\chi_{483}(151,\cdot)\) \(\chi_{483}(163,\cdot)\) \(\chi_{483}(193,\cdot)\) \(\chi_{483}(256,\cdot)\) \(\chi_{483}(289,\cdot)\) \(\chi_{483}(331,\cdot)\) \(\chi_{483}(340,\cdot)\) \(\chi_{483}(361,\cdot)\) \(\chi_{483}(394,\cdot)\) \(\chi_{483}(403,\cdot)\) \(\chi_{483}(445,\cdot)\) \(\chi_{483}(466,\cdot)\) \(\chi_{483}(478,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | 33.33.277966181338944111003326058293667039541136678070715028736001.1 |
Values on generators
\((323,346,442)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{2}{11}\right))\)
Values
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\(1\) | \(1\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{483}(4,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(4,r) e\left(\frac{2r}{483}\right) = -5.8904074034+11.2384652254i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{483}(4,\cdot),\chi_{483}(1,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(4,r) \chi_{483}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{483}(4,·))
= \sum_{r \in \Z/483\Z}
\chi_{483}(4,r) e\left(\frac{1 r + 2 r^{-1}}{483}\right)
= 7.3496719946+-10.3211701533i \)