from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(345, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,36]))
pari: [g,chi] = znchar(Mod(98,345))
Basic properties
Modulus: | \(345\) | |
Conductor: | \(345\) | 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 345.x
\(\chi_{345}(2,\cdot)\) \(\chi_{345}(8,\cdot)\) \(\chi_{345}(32,\cdot)\) \(\chi_{345}(62,\cdot)\) \(\chi_{345}(77,\cdot)\) \(\chi_{345}(98,\cdot)\) \(\chi_{345}(128,\cdot)\) \(\chi_{345}(167,\cdot)\) \(\chi_{345}(173,\cdot)\) \(\chi_{345}(188,\cdot)\) \(\chi_{345}(197,\cdot)\) \(\chi_{345}(233,\cdot)\) \(\chi_{345}(242,\cdot)\) \(\chi_{345}(248,\cdot)\) \(\chi_{345}(257,\cdot)\) \(\chi_{345}(278,\cdot)\) \(\chi_{345}(302,\cdot)\) \(\chi_{345}(308,\cdot)\) \(\chi_{345}(317,\cdot)\) \(\chi_{345}(338,\cdot)\)
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
\((116,277,166)\) → \((-1,-i,e\left(\frac{9}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 345 }(98, a) \) | \(1\) | \(1\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)