from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6864, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,30,24,35]))
pari: [g,chi] = znchar(Mod(401,6864))
Basic properties
Modulus: | \(6864\) | |
Conductor: | \(429\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{429}(401,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6864.lk
\(\chi_{6864}(401,\cdot)\) \(\chi_{6864}(449,\cdot)\) \(\chi_{6864}(929,\cdot)\) \(\chi_{6864}(977,\cdot)\) \(\chi_{6864}(1697,\cdot)\) \(\chi_{6864}(2225,\cdot)\) \(\chi_{6864}(2897,\cdot)\) \(\chi_{6864}(3425,\cdot)\) \(\chi_{6864}(3569,\cdot)\) \(\chi_{6864}(4097,\cdot)\) \(\chi_{6864}(4145,\cdot)\) \(\chi_{6864}(4673,\cdot)\) \(\chi_{6864}(5393,\cdot)\) \(\chi_{6864}(5921,\cdot)\) \(\chi_{6864}(6065,\cdot)\) \(\chi_{6864}(6593,\cdot)\)
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
\((2575,1717,4577,4369,2641)\) → \((1,1,-1,e\left(\frac{2}{5}\right),e\left(\frac{7}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
\( \chi_{ 6864 }(401, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{53}{60}\right)\) |
sage: chi.jacobi_sum(n)