from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4030, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,25,4]))
pari: [g,chi] = znchar(Mod(1011,4030))
Basic properties
Modulus: | \(4030\) | |
Conductor: | \(403\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{403}(205,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4030.fr
\(\chi_{4030}(231,\cdot)\) \(\chi_{4030}(381,\cdot)\) \(\chi_{4030}(751,\cdot)\) \(\chi_{4030}(1011,\cdot)\) \(\chi_{4030}(1291,\cdot)\) \(\chi_{4030}(2591,\cdot)\) \(\chi_{4030}(3221,\cdot)\) \(\chi_{4030}(3761,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | 30.30.4043069762060388435860383865333476800658978190634895286990232354626413.1 |
Values on generators
\((807,1861,2731)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{2}{15}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 4030 }(1011, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) |
sage: chi.jacobi_sum(n)