from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7623, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,0,51]))
pari: [g,chi] = znchar(Mod(5279,7623))
Basic properties
Modulus: | \(7623\) | |
Conductor: | \(1089\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1089}(923,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7623.eh
\(\chi_{7623}(428,\cdot)\) \(\chi_{7623}(659,\cdot)\) \(\chi_{7623}(1121,\cdot)\) \(\chi_{7623}(1352,\cdot)\) \(\chi_{7623}(2045,\cdot)\) \(\chi_{7623}(2507,\cdot)\) \(\chi_{7623}(2738,\cdot)\) \(\chi_{7623}(3200,\cdot)\) \(\chi_{7623}(3431,\cdot)\) \(\chi_{7623}(3893,\cdot)\) \(\chi_{7623}(4124,\cdot)\) \(\chi_{7623}(4586,\cdot)\) \(\chi_{7623}(4817,\cdot)\) \(\chi_{7623}(5279,\cdot)\) \(\chi_{7623}(5510,\cdot)\) \(\chi_{7623}(5972,\cdot)\) \(\chi_{7623}(6203,\cdot)\) \(\chi_{7623}(6665,\cdot)\) \(\chi_{7623}(7358,\cdot)\) \(\chi_{7623}(7589,\cdot)\)
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
\((848,4357,4600)\) → \((e\left(\frac{5}{6}\right),1,e\left(\frac{17}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 7623 }(5279, a) \) | \(1\) | \(1\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{37}{66}\right)\) |
sage: chi.jacobi_sum(n)