sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(276, base_ring=CyclotomicField(22))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,11,2]))
pari: [g,chi] = znchar(Mod(209,276))
Basic properties
| Modulus: | \(276\) | |
| Conductor: | \(69\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{69}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 276.n
\(\chi_{276}(29,\cdot)\) \(\chi_{276}(41,\cdot)\) \(\chi_{276}(77,\cdot)\) \(\chi_{276}(101,\cdot)\) \(\chi_{276}(173,\cdot)\) \(\chi_{276}(197,\cdot)\) \(\chi_{276}(209,\cdot)\) \(\chi_{276}(233,\cdot)\) \(\chi_{276}(257,\cdot)\) \(\chi_{276}(269,\cdot)\)
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: | 22.0.304011857053427966889939263171547.1 |
Values on generators
\((139,185,97)\) → \((1,-1,e\left(\frac{1}{11}\right))\)
Values
| \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \(-1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{276}(209,\cdot)) = \sum_{r\in \Z/276\Z} \chi_{276}(209,r) e\left(\frac{r}{138}\right) = -12.1048450338+-11.378608294i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{276}(209,\cdot),\chi_{276}(1,\cdot)) = \sum_{r\in \Z/276\Z} \chi_{276}(209,r) \chi_{276}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{276}(209,·))
= \sum_{r \in \Z/276\Z}
\chi_{276}(209,r) e\left(\frac{1 r + 2 r^{-1}}{276}\right)
= 0.0 \)