sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(231, base_ring=CyclotomicField(30))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,10,6]))
pari: [g,chi] = znchar(Mod(37,231))
Basic properties
| Modulus: | \(231\) | |
| Conductor: | \(77\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{77}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 231.y
\(\chi_{231}(4,\cdot)\) \(\chi_{231}(16,\cdot)\) \(\chi_{231}(25,\cdot)\) \(\chi_{231}(37,\cdot)\) \(\chi_{231}(58,\cdot)\) \(\chi_{231}(130,\cdot)\) \(\chi_{231}(163,\cdot)\) \(\chi_{231}(214,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 15.15.886528337182930278529.1 |
Values on generators
\((155,199,211)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{1}{5}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{231}(37,\cdot)) = \sum_{r\in \Z/231\Z} \chi_{231}(37,r) e\left(\frac{2r}{231}\right) = -4.2532865662+7.6752559166i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{231}(37,\cdot),\chi_{231}(1,\cdot)) = \sum_{r\in \Z/231\Z} \chi_{231}(37,r) \chi_{231}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{231}(37,·))
= \sum_{r \in \Z/231\Z}
\chi_{231}(37,r) e\left(\frac{1 r + 2 r^{-1}}{231}\right)
= -8.6780619443+3.8637221105i \)