from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(735, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,40]))
pari: [g,chi] = znchar(Mod(11,735))
Basic properties
| Modulus: | \(735\) | |
| Conductor: | \(147\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{147}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 735.bl
\(\chi_{735}(11,\cdot)\) \(\chi_{735}(86,\cdot)\) \(\chi_{735}(191,\cdot)\) \(\chi_{735}(221,\cdot)\) \(\chi_{735}(296,\cdot)\) \(\chi_{735}(326,\cdot)\) \(\chi_{735}(401,\cdot)\) \(\chi_{735}(431,\cdot)\) \(\chi_{735}(506,\cdot)\) \(\chi_{735}(536,\cdot)\) \(\chi_{735}(611,\cdot)\) \(\chi_{735}(641,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.0.176602720807616761537805583365440112858555316650025456145851095351761290003.1 |
Values on generators
\((491,442,346)\) → \((-1,1,e\left(\frac{20}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 735 }(11, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{29}{42}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)