from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7623, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,33,39]))
pari: [g,chi] = znchar(Mod(5389,7623))
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.du
\(\chi_{7623}(76,\cdot)\) \(\chi_{7623}(538,\cdot)\) \(\chi_{7623}(769,\cdot)\) \(\chi_{7623}(1231,\cdot)\) \(\chi_{7623}(1462,\cdot)\) \(\chi_{7623}(1924,\cdot)\) \(\chi_{7623}(2155,\cdot)\) \(\chi_{7623}(2617,\cdot)\) \(\chi_{7623}(2848,\cdot)\) \(\chi_{7623}(3310,\cdot)\) \(\chi_{7623}(3541,\cdot)\) \(\chi_{7623}(4003,\cdot)\) \(\chi_{7623}(4696,\cdot)\) \(\chi_{7623}(4927,\cdot)\) \(\chi_{7623}(5389,\cdot)\) \(\chi_{7623}(5620,\cdot)\) \(\chi_{7623}(6082,\cdot)\) \(\chi_{7623}(6313,\cdot)\) \(\chi_{7623}(7006,\cdot)\) \(\chi_{7623}(7468,\cdot)\)
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
\((848,4357,4600)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{13}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 7623 }(5389, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{5}{66}\right)\) |
sage: chi.jacobi_sum(n)