from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7623, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,44,57]))
pari: [g,chi] = znchar(Mod(2804,7623))
Basic properties
Modulus: | \(7623\) | |
Conductor: | \(7623\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | 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 7623.em
\(\chi_{7623}(32,\cdot)\) \(\chi_{7623}(65,\cdot)\) \(\chi_{7623}(758,\cdot)\) \(\chi_{7623}(1418,\cdot)\) \(\chi_{7623}(2111,\cdot)\) \(\chi_{7623}(2144,\cdot)\) \(\chi_{7623}(2804,\cdot)\) \(\chi_{7623}(2837,\cdot)\) \(\chi_{7623}(3497,\cdot)\) \(\chi_{7623}(3530,\cdot)\) \(\chi_{7623}(4190,\cdot)\) \(\chi_{7623}(4223,\cdot)\) \(\chi_{7623}(4883,\cdot)\) \(\chi_{7623}(4916,\cdot)\) \(\chi_{7623}(5576,\cdot)\) \(\chi_{7623}(5609,\cdot)\) \(\chi_{7623}(6269,\cdot)\) \(\chi_{7623}(6302,\cdot)\) \(\chi_{7623}(6962,\cdot)\) \(\chi_{7623}(6995,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((848,4357,4600)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{2}{3}\right),e\left(\frac{19}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 7623 }(2804, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{31}{66}\right)\) |
sage: chi.jacobi_sum(n)