from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(392, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,10]))
pari: [g,chi] = znchar(Mod(53,392))
Basic properties
Modulus: | \(392\) | |
Conductor: | \(392\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 392.z
\(\chi_{392}(37,\cdot)\) \(\chi_{392}(53,\cdot)\) \(\chi_{392}(93,\cdot)\) \(\chi_{392}(109,\cdot)\) \(\chi_{392}(149,\cdot)\) \(\chi_{392}(205,\cdot)\) \(\chi_{392}(221,\cdot)\) \(\chi_{392}(261,\cdot)\) \(\chi_{392}(277,\cdot)\) \(\chi_{392}(317,\cdot)\) \(\chi_{392}(333,\cdot)\) \(\chi_{392}(389,\cdot)\)
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: | 42.42.155718699466313184257207094263668545441599708733396657696588937331033553383727300608.1 |
Values on generators
\((295,197,297)\) → \((1,-1,e\left(\frac{5}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 392 }(53, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) |
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)