from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9075, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,20]))
pari: [g,chi] = znchar(Mod(518,9075))
Basic properties
Modulus: | \(9075\) | |
Conductor: | \(1815\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1815}(518,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9075.dp
\(\chi_{9075}(518,\cdot)\) \(\chi_{9075}(782,\cdot)\) \(\chi_{9075}(1343,\cdot)\) \(\chi_{9075}(1607,\cdot)\) \(\chi_{9075}(2168,\cdot)\) \(\chi_{9075}(2432,\cdot)\) \(\chi_{9075}(2993,\cdot)\) \(\chi_{9075}(3257,\cdot)\) \(\chi_{9075}(3818,\cdot)\) \(\chi_{9075}(4082,\cdot)\) \(\chi_{9075}(4643,\cdot)\) \(\chi_{9075}(4907,\cdot)\) \(\chi_{9075}(5468,\cdot)\) \(\chi_{9075}(5732,\cdot)\) \(\chi_{9075}(6557,\cdot)\) \(\chi_{9075}(7118,\cdot)\) \(\chi_{9075}(7943,\cdot)\) \(\chi_{9075}(8207,\cdot)\) \(\chi_{9075}(8768,\cdot)\) \(\chi_{9075}(9032,\cdot)\)
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
\((3026,727,5326)\) → \((-1,-i,e\left(\frac{5}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) | \(23\) |
\( \chi_{ 9075 }(518, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{25}{44}\right)\) |
sage: chi.jacobi_sum(n)