from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1150, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([11,38]))
pari: [g,chi] = znchar(Mod(7,1150))
Basic properties
| Modulus: | \(1150\) | |
| 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: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1150.r
\(\chi_{1150}(7,\cdot)\) \(\chi_{1150}(43,\cdot)\) \(\chi_{1150}(57,\cdot)\) \(\chi_{1150}(107,\cdot)\) \(\chi_{1150}(143,\cdot)\) \(\chi_{1150}(157,\cdot)\) \(\chi_{1150}(293,\cdot)\) \(\chi_{1150}(343,\cdot)\) \(\chi_{1150}(457,\cdot)\) \(\chi_{1150}(493,\cdot)\) \(\chi_{1150}(543,\cdot)\) \(\chi_{1150}(557,\cdot)\) \(\chi_{1150}(707,\cdot)\) \(\chi_{1150}(743,\cdot)\) \(\chi_{1150}(757,\cdot)\) \(\chi_{1150}(793,\cdot)\) \(\chi_{1150}(843,\cdot)\) \(\chi_{1150}(893,\cdot)\) \(\chi_{1150}(907,\cdot)\) \(\chi_{1150}(957,\cdot)\)
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
\((277,51)\) → \((i,e\left(\frac{19}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 1150 }(7, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) |
sage: chi.jacobi_sum(n)