from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4290, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,15,12,35]))
pari: [g,chi] = znchar(Mod(37,4290))
Basic properties
Modulus: | \(4290\) | |
Conductor: | \(715\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{715}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4290.gb
\(\chi_{4290}(37,\cdot)\) \(\chi_{4290}(223,\cdot)\) \(\chi_{4290}(427,\cdot)\) \(\chi_{4290}(643,\cdot)\) \(\chi_{4290}(817,\cdot)\) \(\chi_{4290}(1423,\cdot)\) \(\chi_{4290}(1567,\cdot)\) \(\chi_{4290}(1813,\cdot)\) \(\chi_{4290}(2203,\cdot)\) \(\chi_{4290}(2347,\cdot)\) \(\chi_{4290}(2737,\cdot)\) \(\chi_{4290}(2953,\cdot)\) \(\chi_{4290}(3127,\cdot)\) \(\chi_{4290}(3547,\cdot)\) \(\chi_{4290}(3733,\cdot)\) \(\chi_{4290}(4123,\cdot)\)
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
\((2861,1717,3511,2641)\) → \((1,i,e\left(\frac{1}{5}\right),e\left(\frac{7}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 4290 }(37, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{3}{5}\right)\) |
sage: chi.jacobi_sum(n)