from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4030, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,10,46]))
pari: [g,chi] = znchar(Mod(693,4030))
Basic properties
Modulus: | \(4030\) | |
Conductor: | \(2015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2015}(693,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4030.gm
\(\chi_{4030}(127,\cdot)\) \(\chi_{4030}(693,\cdot)\) \(\chi_{4030}(797,\cdot)\) \(\chi_{4030}(933,\cdot)\) \(\chi_{4030}(1447,\cdot)\) \(\chi_{4030}(1603,\cdot)\) \(\chi_{4030}(2253,\cdot)\) \(\chi_{4030}(2337,\cdot)\) \(\chi_{4030}(2597,\cdot)\) \(\chi_{4030}(2617,\cdot)\) \(\chi_{4030}(3117,\cdot)\) \(\chi_{4030}(3143,\cdot)\) \(\chi_{4030}(3403,\cdot)\) \(\chi_{4030}(3423,\cdot)\) \(\chi_{4030}(3917,\cdot)\) \(\chi_{4030}(3923,\cdot)\)
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
\((807,1861,2731)\) → \((-i,e\left(\frac{1}{6}\right),e\left(\frac{23}{30}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 4030 }(693, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) |
sage: chi.jacobi_sum(n)