sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(245, base_ring=CyclotomicField(14))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,6]))
pari: [g,chi] = znchar(Mod(176,245))
Basic properties
| Modulus: | \(245\) | |
| Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(7\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{49}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 245.k
\(\chi_{245}(36,\cdot)\) \(\chi_{245}(71,\cdot)\) \(\chi_{245}(106,\cdot)\) \(\chi_{245}(141,\cdot)\) \(\chi_{245}(176,\cdot)\) \(\chi_{245}(211,\cdot)\)
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: | \(\Q(\zeta_{7})\) |
| Fixed field: | 7.7.13841287201.1 |
Values on generators
\((197,101)\) → \((1,e\left(\frac{3}{7}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \(1\) | \(1\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{245}(176,\cdot)) = \sum_{r\in \Z/245\Z} \chi_{245}(176,r) e\left(\frac{2r}{245}\right) = 4.0048166209+5.7412057822i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{245}(176,\cdot),\chi_{245}(1,\cdot)) = \sum_{r\in \Z/245\Z} \chi_{245}(176,r) \chi_{245}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{245}(176,·))
= \sum_{r \in \Z/245\Z}
\chi_{245}(176,r) e\left(\frac{1 r + 2 r^{-1}}{245}\right)
= 39.483381269+19.0141942839i \)