from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6045, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,25,4]))
pari: [g,chi] = znchar(Mod(71,6045))
Basic properties
Modulus: | \(6045\) | |
Conductor: | \(1209\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1209}(71,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6045.my
\(\chi_{6045}(71,\cdot)\) \(\chi_{6045}(236,\cdot)\) \(\chi_{6045}(266,\cdot)\) \(\chi_{6045}(431,\cdot)\) \(\chi_{6045}(596,\cdot)\) \(\chi_{6045}(1436,\cdot)\) \(\chi_{6045}(1931,\cdot)\) \(\chi_{6045}(1991,\cdot)\) \(\chi_{6045}(2126,\cdot)\) \(\chi_{6045}(2996,\cdot)\) \(\chi_{6045}(3296,\cdot)\) \(\chi_{6045}(4106,\cdot)\) \(\chi_{6045}(4856,\cdot)\) \(\chi_{6045}(4886,\cdot)\) \(\chi_{6045}(5081,\cdot)\) \(\chi_{6045}(5501,\cdot)\)
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
\((4031,4837,1861,2731)\) → \((-1,1,e\left(\frac{5}{12}\right),e\left(\frac{1}{15}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
\( \chi_{ 6045 }(71, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) |
sage: chi.jacobi_sum(n)