from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3450, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,38]))
pari: [g,chi] = znchar(Mod(7,3450))
Basic properties
Modulus: | \(3450\) | |
Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{115}(7,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3450.bi
\(\chi_{3450}(7,\cdot)\) \(\chi_{3450}(43,\cdot)\) \(\chi_{3450}(157,\cdot)\) \(\chi_{3450}(343,\cdot)\) \(\chi_{3450}(457,\cdot)\) \(\chi_{3450}(493,\cdot)\) \(\chi_{3450}(757,\cdot)\) \(\chi_{3450}(793,\cdot)\) \(\chi_{3450}(907,\cdot)\) \(\chi_{3450}(1207,\cdot)\) \(\chi_{3450}(1693,\cdot)\) \(\chi_{3450}(1993,\cdot)\) \(\chi_{3450}(2107,\cdot)\) \(\chi_{3450}(2407,\cdot)\) \(\chi_{3450}(2443,\cdot)\) \(\chi_{3450}(2593,\cdot)\) \(\chi_{3450}(2857,\cdot)\) \(\chi_{3450}(3007,\cdot)\) \(\chi_{3450}(3043,\cdot)\) \(\chi_{3450}(3193,\cdot)\)
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: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((1151,277,1201)\) → \((1,i,e\left(\frac{19}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 3450 }(7, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) |
sage: chi.jacobi_sum(n)