from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6008, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,0,26]))
pari: [g,chi] = znchar(Mod(1201,6008))
Basic properties
Modulus: | \(6008\) | |
Conductor: | \(751\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(25\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{751}(450,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6008.z
\(\chi_{6008}(193,\cdot)\) \(\chi_{6008}(481,\cdot)\) \(\chi_{6008}(913,\cdot)\) \(\chi_{6008}(1201,\cdot)\) \(\chi_{6008}(1217,\cdot)\) \(\chi_{6008}(1417,\cdot)\) \(\chi_{6008}(1553,\cdot)\) \(\chi_{6008}(1673,\cdot)\) \(\chi_{6008}(1681,\cdot)\) \(\chi_{6008}(1977,\cdot)\) \(\chi_{6008}(2001,\cdot)\) \(\chi_{6008}(2601,\cdot)\) \(\chi_{6008}(2673,\cdot)\) \(\chi_{6008}(2809,\cdot)\) \(\chi_{6008}(3057,\cdot)\) \(\chi_{6008}(3121,\cdot)\) \(\chi_{6008}(3329,\cdot)\) \(\chi_{6008}(3489,\cdot)\) \(\chi_{6008}(4465,\cdot)\) \(\chi_{6008}(5209,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
Fixed field: | Number field defined by a degree 25 polynomial |
Values on generators
\((1503,3005,1505)\) → \((1,1,e\left(\frac{13}{25}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 6008 }(1201, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) |
sage: chi.jacobi_sum(n)