from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7623, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,44,42]))
pari: [g,chi] = znchar(Mod(6073,7623))
Basic properties
Modulus: | \(7623\) | |
Conductor: | \(7623\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | 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.dk
\(\chi_{7623}(529,\cdot)\) \(\chi_{7623}(562,\cdot)\) \(\chi_{7623}(1222,\cdot)\) \(\chi_{7623}(1255,\cdot)\) \(\chi_{7623}(1915,\cdot)\) \(\chi_{7623}(1948,\cdot)\) \(\chi_{7623}(2608,\cdot)\) \(\chi_{7623}(2641,\cdot)\) \(\chi_{7623}(3301,\cdot)\) \(\chi_{7623}(3334,\cdot)\) \(\chi_{7623}(4027,\cdot)\) \(\chi_{7623}(4687,\cdot)\) \(\chi_{7623}(5380,\cdot)\) \(\chi_{7623}(5413,\cdot)\) \(\chi_{7623}(6073,\cdot)\) \(\chi_{7623}(6106,\cdot)\) \(\chi_{7623}(6766,\cdot)\) \(\chi_{7623}(6799,\cdot)\) \(\chi_{7623}(7459,\cdot)\) \(\chi_{7623}(7492,\cdot)\)
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 33 polynomial |
Values on generators
\((848,4357,4600)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{2}{3}\right),e\left(\frac{7}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 7623 }(6073, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) |
sage: chi.jacobi_sum(n)