from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4030, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,55,14]))
pari: [g,chi] = znchar(Mod(761,4030))
Basic properties
Modulus: | \(4030\) | |
Conductor: | \(403\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{403}(358,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4030.hq
\(\chi_{4030}(11,\cdot)\) \(\chi_{4030}(241,\cdot)\) \(\chi_{4030}(301,\cdot)\) \(\chi_{4030}(761,\cdot)\) \(\chi_{4030}(921,\cdot)\) \(\chi_{4030}(951,\cdot)\) \(\chi_{4030}(1571,\cdot)\) \(\chi_{4030}(1801,\cdot)\) \(\chi_{4030}(2121,\cdot)\) \(\chi_{4030}(2151,\cdot)\) \(\chi_{4030}(2411,\cdot)\) \(\chi_{4030}(2741,\cdot)\) \(\chi_{4030}(2931,\cdot)\) \(\chi_{4030}(3421,\cdot)\) \(\chi_{4030}(3971,\cdot)\) \(\chi_{4030}(4011,\cdot)\)
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
\((807,1861,2731)\) → \((1,e\left(\frac{11}{12}\right),e\left(\frac{7}{30}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 4030 }(761, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{23}{30}\right)\) |
sage: chi.jacobi_sum(n)