from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2300, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,11,38]))
pari: [g,chi] = znchar(Mod(7,2300))
Basic properties
| Modulus: | \(2300\) | |
| Conductor: | \(460\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{460}(7,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2300.bh
\(\chi_{2300}(7,\cdot)\) \(\chi_{2300}(43,\cdot)\) \(\chi_{2300}(107,\cdot)\) \(\chi_{2300}(143,\cdot)\) \(\chi_{2300}(343,\cdot)\) \(\chi_{2300}(543,\cdot)\) \(\chi_{2300}(707,\cdot)\) \(\chi_{2300}(743,\cdot)\) \(\chi_{2300}(843,\cdot)\) \(\chi_{2300}(907,\cdot)\) \(\chi_{2300}(1207,\cdot)\) \(\chi_{2300}(1307,\cdot)\) \(\chi_{2300}(1443,\cdot)\) \(\chi_{2300}(1607,\cdot)\) \(\chi_{2300}(1643,\cdot)\) \(\chi_{2300}(1707,\cdot)\) \(\chi_{2300}(1907,\cdot)\) \(\chi_{2300}(1943,\cdot)\) \(\chi_{2300}(2043,\cdot)\) \(\chi_{2300}(2107,\cdot)\)
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: | 44.0.3190796191738142789235043789002363949895144644980550209800192000000000000000000000000000000000.1 |
Values on generators
\((1151,277,1201)\) → \((-1,i,e\left(\frac{19}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 2300 }(7, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) |
sage: chi.jacobi_sum(n)