from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7623, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([11,44,51]))
pari: [g,chi] = znchar(Mod(1649,7623))
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.dq
\(\chi_{7623}(263,\cdot)\) \(\chi_{7623}(527,\cdot)\) \(\chi_{7623}(956,\cdot)\) \(\chi_{7623}(1220,\cdot)\) \(\chi_{7623}(1649,\cdot)\) \(\chi_{7623}(1913,\cdot)\) \(\chi_{7623}(2342,\cdot)\) \(\chi_{7623}(2606,\cdot)\) \(\chi_{7623}(3035,\cdot)\) \(\chi_{7623}(3299,\cdot)\) \(\chi_{7623}(3728,\cdot)\) \(\chi_{7623}(4421,\cdot)\) \(\chi_{7623}(4685,\cdot)\) \(\chi_{7623}(5114,\cdot)\) \(\chi_{7623}(5378,\cdot)\) \(\chi_{7623}(6071,\cdot)\) \(\chi_{7623}(6500,\cdot)\) \(\chi_{7623}(6764,\cdot)\) \(\chi_{7623}(7193,\cdot)\) \(\chi_{7623}(7457,\cdot)\)
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
\((848,4357,4600)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{2}{3}\right),e\left(\frac{17}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 7623 }(1649, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{59}{66}\right)\) |
sage: chi.jacobi_sum(n)