from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4704, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,0,17]))
pari: [g,chi] = znchar(Mod(271,4704))
Basic properties
Modulus: | \(4704\) | |
Conductor: | \(392\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{392}(75,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4704.dx
\(\chi_{4704}(271,\cdot)\) \(\chi_{4704}(367,\cdot)\) \(\chi_{4704}(943,\cdot)\) \(\chi_{4704}(1039,\cdot)\) \(\chi_{4704}(1615,\cdot)\) \(\chi_{4704}(1711,\cdot)\) \(\chi_{4704}(2287,\cdot)\) \(\chi_{4704}(3055,\cdot)\) \(\chi_{4704}(3631,\cdot)\) \(\chi_{4704}(3727,\cdot)\) \(\chi_{4704}(4303,\cdot)\) \(\chi_{4704}(4399,\cdot)\)
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: | 42.42.1090030896264192289800449659845679818091197961133776603876122561317234873686091104256.1 |
Values on generators
\((1471,1765,3137,4609)\) → \((-1,-1,1,e\left(\frac{17}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 4704 }(271, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{19}{42}\right)\) |
sage: chi.jacobi_sum(n)