from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1815, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,2]))
pari: [g,chi] = znchar(Mod(1363,1815))
Basic properties
| Modulus: | \(1815\) | |
| Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{605}(153,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1815.bi
\(\chi_{1815}(43,\cdot)\) \(\chi_{1815}(142,\cdot)\) \(\chi_{1815}(208,\cdot)\) \(\chi_{1815}(307,\cdot)\) \(\chi_{1815}(373,\cdot)\) \(\chi_{1815}(472,\cdot)\) \(\chi_{1815}(538,\cdot)\) \(\chi_{1815}(637,\cdot)\) \(\chi_{1815}(703,\cdot)\) \(\chi_{1815}(802,\cdot)\) \(\chi_{1815}(868,\cdot)\) \(\chi_{1815}(1033,\cdot)\) \(\chi_{1815}(1132,\cdot)\) \(\chi_{1815}(1198,\cdot)\) \(\chi_{1815}(1297,\cdot)\) \(\chi_{1815}(1363,\cdot)\) \(\chi_{1815}(1462,\cdot)\) \(\chi_{1815}(1528,\cdot)\) \(\chi_{1815}(1627,\cdot)\) \(\chi_{1815}(1792,\cdot)\)
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: | 44.44.2885428559557085084648615903962269104974580506944665166312236845353556846511909399754484184086322784423828125.1 |
Values on generators
\((1211,727,1696)\) → \((1,-i,e\left(\frac{1}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
| \( \chi_{ 1815 }(1363, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) |
sage: chi.jacobi_sum(n)