from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6864, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,15,10,2,0]))
pari: [g,chi] = znchar(Mod(365,6864))
Basic properties
Modulus: | \(6864\) | |
Conductor: | \(528\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{528}(365,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6864.hv
\(\chi_{6864}(365,\cdot)\) \(\chi_{6864}(677,\cdot)\) \(\chi_{6864}(1613,\cdot)\) \(\chi_{6864}(2549,\cdot)\) \(\chi_{6864}(3797,\cdot)\) \(\chi_{6864}(4109,\cdot)\) \(\chi_{6864}(5045,\cdot)\) \(\chi_{6864}(5981,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | 20.20.11828526346956240442271551059351979425792.1 |
Values on generators
\((2575,1717,4577,4369,2641)\) → \((1,-i,-1,e\left(\frac{1}{10}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
\( \chi_{ 6864 }(365, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) |
sage: chi.jacobi_sum(n)