from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3450, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,10]))
pari: [g,chi] = znchar(Mod(1193,3450))
Basic properties
Modulus: | \(3450\) | |
Conductor: | \(345\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{345}(158,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3450.bg
\(\chi_{3450}(107,\cdot)\) \(\chi_{3450}(143,\cdot)\) \(\chi_{3450}(293,\cdot)\) \(\chi_{3450}(557,\cdot)\) \(\chi_{3450}(707,\cdot)\) \(\chi_{3450}(743,\cdot)\) \(\chi_{3450}(893,\cdot)\) \(\chi_{3450}(1157,\cdot)\) \(\chi_{3450}(1193,\cdot)\) \(\chi_{3450}(1307,\cdot)\) \(\chi_{3450}(1493,\cdot)\) \(\chi_{3450}(1607,\cdot)\) \(\chi_{3450}(1643,\cdot)\) \(\chi_{3450}(1907,\cdot)\) \(\chi_{3450}(1943,\cdot)\) \(\chi_{3450}(2057,\cdot)\) \(\chi_{3450}(2357,\cdot)\) \(\chi_{3450}(2843,\cdot)\) \(\chi_{3450}(3143,\cdot)\) \(\chi_{3450}(3257,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((1151,277,1201)\) → \((-1,-i,e\left(\frac{5}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 3450 }(1193, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{17}{44}\right)\) |
sage: chi.jacobi_sum(n)