from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3150, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([50,57,0]))
pari: [g,chi] = znchar(Mod(113,3150))
Basic properties
Modulus: | \(3150\) | |
Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{225}(113,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3150.eh
\(\chi_{3150}(113,\cdot)\) \(\chi_{3150}(533,\cdot)\) \(\chi_{3150}(617,\cdot)\) \(\chi_{3150}(1037,\cdot)\) \(\chi_{3150}(1163,\cdot)\) \(\chi_{3150}(1247,\cdot)\) \(\chi_{3150}(1373,\cdot)\) \(\chi_{3150}(1667,\cdot)\) \(\chi_{3150}(1877,\cdot)\) \(\chi_{3150}(2003,\cdot)\) \(\chi_{3150}(2297,\cdot)\) \(\chi_{3150}(2423,\cdot)\) \(\chi_{3150}(2633,\cdot)\) \(\chi_{3150}(2927,\cdot)\) \(\chi_{3150}(3053,\cdot)\) \(\chi_{3150}(3137,\cdot)\)
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
\((2801,127,451)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{19}{20}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 3150 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)