from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(980, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,16]))
pari: [g,chi] = znchar(Mod(711,980))
Basic properties
Modulus: | \(980\) | |
Conductor: | \(196\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{196}(123,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 980.bm
\(\chi_{980}(11,\cdot)\) \(\chi_{980}(51,\cdot)\) \(\chi_{980}(151,\cdot)\) \(\chi_{980}(191,\cdot)\) \(\chi_{980}(291,\cdot)\) \(\chi_{980}(331,\cdot)\) \(\chi_{980}(431,\cdot)\) \(\chi_{980}(571,\cdot)\) \(\chi_{980}(611,\cdot)\) \(\chi_{980}(711,\cdot)\) \(\chi_{980}(751,\cdot)\) \(\chi_{980}(891,\cdot)\)
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: | 42.0.74252462132603256348231837398371002884673933378885582779211491265789772693504.1 |
Values on generators
\((491,197,101)\) → \((-1,1,e\left(\frac{8}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 980 }(711, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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)