from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9200, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,22,33,26]))
pari: [g,chi] = znchar(Mod(343,9200))
Basic properties
| Modulus: | \(9200\) | |
| Conductor: | \(920\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{920}(803,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9200.ee
\(\chi_{9200}(7,\cdot)\) \(\chi_{9200}(343,\cdot)\) \(\chi_{9200}(743,\cdot)\) \(\chi_{9200}(1207,\cdot)\) \(\chi_{9200}(1607,\cdot)\) \(\chi_{9200}(1943,\cdot)\) \(\chi_{9200}(2343,\cdot)\) \(\chi_{9200}(2407,\cdot)\) \(\chi_{9200}(3143,\cdot)\) \(\chi_{9200}(3207,\cdot)\) \(\chi_{9200}(3607,\cdot)\) \(\chi_{9200}(3943,\cdot)\) \(\chi_{9200}(4007,\cdot)\) \(\chi_{9200}(4343,\cdot)\) \(\chi_{9200}(4407,\cdot)\) \(\chi_{9200}(4743,\cdot)\) \(\chi_{9200}(5143,\cdot)\) \(\chi_{9200}(7607,\cdot)\) \(\chi_{9200}(8343,\cdot)\) \(\chi_{9200}(8807,\cdot)\)
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: | 44.0.13383169230192059253459701104387771124501004765020501667165784506368000000000000000000000000000000000.1 |
Values on generators
\((1151,6901,2577,1201)\) → \((-1,-1,-i,e\left(\frac{13}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 9200 }(343, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) |
sage: chi.jacobi_sum(n)