from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6720, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,33,0,12,8]))
pari: [g,chi] = znchar(Mod(157,6720))
Basic properties
| Modulus: | \(6720\) | |
| Conductor: | \(2240\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2240}(157,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6720.jw
\(\chi_{6720}(157,\cdot)\) \(\chi_{6720}(397,\cdot)\) \(\chi_{6720}(1333,\cdot)\) \(\chi_{6720}(1573,\cdot)\) \(\chi_{6720}(1837,\cdot)\) \(\chi_{6720}(2077,\cdot)\) \(\chi_{6720}(3013,\cdot)\) \(\chi_{6720}(3253,\cdot)\) \(\chi_{6720}(3517,\cdot)\) \(\chi_{6720}(3757,\cdot)\) \(\chi_{6720}(4693,\cdot)\) \(\chi_{6720}(4933,\cdot)\) \(\chi_{6720}(5197,\cdot)\) \(\chi_{6720}(5437,\cdot)\) \(\chi_{6720}(6373,\cdot)\) \(\chi_{6720}(6613,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((1471,3781,4481,5377,1921)\) → \((1,e\left(\frac{11}{16}\right),1,i,e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6720 }(157, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) |
sage: chi.jacobi_sum(n)