from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(172, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,26]))
pari: [g,chi] = znchar(Mod(15,172))
Basic properties
Modulus: | \(172\) | |
Conductor: | \(172\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 172.o
\(\chi_{172}(15,\cdot)\) \(\chi_{172}(23,\cdot)\) \(\chi_{172}(31,\cdot)\) \(\chi_{172}(67,\cdot)\) \(\chi_{172}(83,\cdot)\) \(\chi_{172}(95,\cdot)\) \(\chi_{172}(99,\cdot)\) \(\chi_{172}(103,\cdot)\) \(\chi_{172}(111,\cdot)\) \(\chi_{172}(139,\cdot)\) \(\chi_{172}(143,\cdot)\) \(\chi_{172}(167,\cdot)\)
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: | 42.0.959396304051793463814262846982490027578741814649477038563926538598268329263104.1 |
Values on generators
\((87,89)\) → \((-1,e\left(\frac{13}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 172 }(15, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) |
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)