from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9200, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,0,20]))
pari: [g,chi] = znchar(Mod(101,9200))
Basic properties
| Modulus: | \(9200\) | |
| Conductor: | \(368\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{368}(101,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9200.ej
\(\chi_{9200}(101,\cdot)\) \(\chi_{9200}(301,\cdot)\) \(\chi_{9200}(501,\cdot)\) \(\chi_{9200}(901,\cdot)\) \(\chi_{9200}(1301,\cdot)\) \(\chi_{9200}(1501,\cdot)\) \(\chi_{9200}(2101,\cdot)\) \(\chi_{9200}(2901,\cdot)\) \(\chi_{9200}(3301,\cdot)\) \(\chi_{9200}(4501,\cdot)\) \(\chi_{9200}(4701,\cdot)\) \(\chi_{9200}(4901,\cdot)\) \(\chi_{9200}(5101,\cdot)\) \(\chi_{9200}(5501,\cdot)\) \(\chi_{9200}(5901,\cdot)\) \(\chi_{9200}(6101,\cdot)\) \(\chi_{9200}(6701,\cdot)\) \(\chi_{9200}(7501,\cdot)\) \(\chi_{9200}(7901,\cdot)\) \(\chi_{9200}(9101,\cdot)\)
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: | 44.44.7829660228065619245582194641412012312544945884150589900838471630076269829766255604192509952.1 |
Values on generators
\((1151,6901,2577,1201)\) → \((1,i,1,e\left(\frac{5}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 9200 }(101, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) |
sage: chi.jacobi_sum(n)