from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4030, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,38]))
pari: [g,chi] = znchar(Mod(291,4030))
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 \(\chi_{403}(291,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4030.gb
\(\chi_{4030}(21,\cdot)\) \(\chi_{4030}(291,\cdot)\) \(\chi_{4030}(551,\cdot)\) \(\chi_{4030}(941,\cdot)\) \(\chi_{4030}(1071,\cdot)\) \(\chi_{4030}(1191,\cdot)\) \(\chi_{4030}(1841,\cdot)\) \(\chi_{4030}(1851,\cdot)\) \(\chi_{4030}(2101,\cdot)\) \(\chi_{4030}(2111,\cdot)\) \(\chi_{4030}(2491,\cdot)\) \(\chi_{4030}(2501,\cdot)\) \(\chi_{4030}(2621,\cdot)\) \(\chi_{4030}(3401,\cdot)\) \(\chi_{4030}(3661,\cdot)\) \(\chi_{4030}(3671,\cdot)\)
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
\((807,1861,2731)\) → \((1,-i,e\left(\frac{19}{30}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 4030 }(291, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) |
sage: chi.jacobi_sum(n)