from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1815, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,11,28]))
pari: [g,chi] = znchar(Mod(1112,1815))
Basic properties
Modulus: | \(1815\) | |
Conductor: | \(1815\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1815.bh
\(\chi_{1815}(23,\cdot)\) \(\chi_{1815}(188,\cdot)\) \(\chi_{1815}(287,\cdot)\) \(\chi_{1815}(353,\cdot)\) \(\chi_{1815}(452,\cdot)\) \(\chi_{1815}(518,\cdot)\) \(\chi_{1815}(617,\cdot)\) \(\chi_{1815}(683,\cdot)\) \(\chi_{1815}(782,\cdot)\) \(\chi_{1815}(947,\cdot)\) \(\chi_{1815}(1013,\cdot)\) \(\chi_{1815}(1112,\cdot)\) \(\chi_{1815}(1178,\cdot)\) \(\chi_{1815}(1277,\cdot)\) \(\chi_{1815}(1343,\cdot)\) \(\chi_{1815}(1442,\cdot)\) \(\chi_{1815}(1508,\cdot)\) \(\chi_{1815}(1607,\cdot)\) \(\chi_{1815}(1673,\cdot)\) \(\chi_{1815}(1772,\cdot)\)
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: | Number field defined by a degree 44 polynomial |
Values on generators
\((1211,727,1696)\) → \((-1,i,e\left(\frac{7}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
\( \chi_{ 1815 }(1112, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{35}{44}\right)\) |
sage: chi.jacobi_sum(n)