from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3344, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,15,24,10]))
pari: [g,chi] = znchar(Mod(27,3344))
Basic properties
Modulus: | \(3344\) | |
Conductor: | \(3344\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | 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 3344.em
\(\chi_{3344}(27,\cdot)\) \(\chi_{3344}(179,\cdot)\) \(\chi_{3344}(411,\cdot)\) \(\chi_{3344}(867,\cdot)\) \(\chi_{3344}(939,\cdot)\) \(\chi_{3344}(1171,\cdot)\) \(\chi_{3344}(1323,\cdot)\) \(\chi_{3344}(1395,\cdot)\) \(\chi_{3344}(1699,\cdot)\) \(\chi_{3344}(1851,\cdot)\) \(\chi_{3344}(2083,\cdot)\) \(\chi_{3344}(2539,\cdot)\) \(\chi_{3344}(2611,\cdot)\) \(\chi_{3344}(2843,\cdot)\) \(\chi_{3344}(2995,\cdot)\) \(\chi_{3344}(3067,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((2927,837,2433,705)\) → \((-1,i,e\left(\frac{2}{5}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
\( \chi_{ 3344 }(27, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{30}\right)\) |
sage: chi.jacobi_sum(n)