from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3104, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,8,25]))
pari: [g,chi] = znchar(Mod(2585,3104))
Basic properties
| Modulus: | \(3104\) | |
| Conductor: | \(1552\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1552}(645,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3104.ee
\(\chi_{3104}(249,\cdot)\) \(\chi_{3104}(633,\cdot)\) \(\chi_{3104}(649,\cdot)\) \(\chi_{3104}(1097,\cdot)\) \(\chi_{3104}(1113,\cdot)\) \(\chi_{3104}(1497,\cdot)\) \(\chi_{3104}(2009,\cdot)\) \(\chi_{3104}(2057,\cdot)\) \(\chi_{3104}(2089,\cdot)\) \(\chi_{3104}(2153,\cdot)\) \(\chi_{3104}(2265,\cdot)\) \(\chi_{3104}(2585,\cdot)\) \(\chi_{3104}(2697,\cdot)\) \(\chi_{3104}(2761,\cdot)\) \(\chi_{3104}(2793,\cdot)\) \(\chi_{3104}(2841,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | 32.0.12038250625011274840538078895284459382192972291498086142512881074076219367002527779258368.1 |
Values on generators
\((2911,389,2721)\) → \((1,i,e\left(\frac{25}{32}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 3104 }(2585, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) |
sage: chi.jacobi_sum(n)